(BOOT-STRAP NQTHM) [ 0.1 0.1 0.0 ] GROUND-ZERO (DEFN BOARDP (X) (IF (NLISTP X) (EQUAL X NIL) (AND (OR (EQUAL (CAR X) 'X) (EQUAL (CAR X) 'O) (EQUAL (CAR X) F)) (BOARDP (CDR X))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to show that the measure (COUNT X) decreases according to the well-founded relation LESSP in each recursive call. Hence, BOARDP is accepted under the principle of definition. From the definition we can conclude that (OR (FALSEP (BOARDP X)) (TRUEP (BOARDP X))) is a theorem. [ 0.0 0.0 0.0 ] BOARDP (DEFN ENDP (BOARD) (IF (NLISTP BOARD) T (IF (CAR BOARD) (ENDP (CDR BOARD)) F))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to show that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Hence, ENDP is accepted under the definitional principle. From the definition we can conclude that (OR (FALSEP (ENDP BOARD)) (TRUEP (ENDP BOARD))) is a theorem. [ 0.0 0.0 0.0 ] ENDP (DEFN LEGAL-MOVEP (BOARD1 PLAYER BOARD2) (COND ((NLISTP BOARD1) F) ((EQUAL (CAR BOARD1) (CAR BOARD2)) (LEGAL-MOVEP (CDR BOARD1) PLAYER (CDR BOARD2))) ((AND (EQUAL (CAR BOARD1) F) (EQUAL (CAR BOARD2) PLAYER)) (EQUAL (CDR BOARD1) (CDR BOARD2))) (T F))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to prove that the measure (COUNT BOARD1) decreases according to the well-founded relation LESSP in each recursive call. Hence, LEGAL-MOVEP is accepted under the principle of definition. Observe that: (OR (FALSEP (LEGAL-MOVEP BOARD1 PLAYER BOARD2)) (TRUEP (LEGAL-MOVEP BOARD1 PLAYER BOARD2))) is a theorem. [ 0.0 0.0 0.0 ] LEGAL-MOVEP (DEFN NTH (I BOARD) (IF (ZEROP I) (CAR BOARD) (NTH (SUB1 I) (CDR BOARD)))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each recursive call. Hence, NTH is accepted under the principle of definition. [ 0.0 0.0 0.0 ] NTH (DEFN WINP1 (X LINE BOARD) (IF (NLISTP LINE) T (IF (EQUAL X (NTH (CAR LINE) BOARD)) (WINP1 X (CDR LINE) BOARD) F))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT LINE) decreases according to the well-founded relation LESSP in each recursive call. Hence, WINP1 is accepted under the principle of definition. From the definition we can conclude that: (OR (FALSEP (WINP1 X LINE BOARD)) (TRUEP (WINP1 X LINE BOARD))) is a theorem. [ 0.0 0.0 0.0 ] WINP1 (DEFN WINP (X BOARD) (OR (WINP1 X '(0 1 2) BOARD) (WINP1 X '(3 4 5) BOARD) (WINP1 X '(6 7 8) BOARD) (WINP1 X '(0 3 6) BOARD) (WINP1 X '(1 4 7) BOARD) (WINP1 X '(2 5 8) BOARD) (WINP1 X '(0 4 8) BOARD) (WINP1 X '(2 4 6) BOARD))) Observe that (OR (FALSEP (WINP X BOARD)) (TRUEP (WINP X BOARD))) is a theorem. [ 0.1 0.0 0.0 ] WINP (DISABLE WINP) [ 0.0 0.0 0.0 ] WINP-OFF (DEFN OTHER-PLAYER (PLAYER) (IF (EQUAL PLAYER 'X) 'O 'X)) Observe that (LITATOM (OTHER-PLAYER PLAYER)) is a theorem. [ 0.0 0.0 0.0 ] OTHER-PLAYER (DEFN END (OUTCOME PLAYER LST) (LET ((OUTCOME (IF (EQUAL OUTCOME 'WIN) (IF (EQUAL PLAYER 'O) 'O-WIN 'X-WIN) 'DRAW))) (IF (EQUAL LST (LIST OUTCOME)) OUTCOME F))) Note that: (OR (FALSEP (END OUTCOME PLAYER LST)) (LITATOM (END OUTCOME PLAYER LST))) is a theorem. [ 0.0 0.0 0.0 ] END (DEFN TIC-TAC-TOEP1 (BOARD PLAYER LST) (COND ((NLISTP LST) (END 'WIN (OTHER-PLAYER PLAYER) LST)) ((NOT (BOARDP (CAR LST))) F) ((NOT (LEGAL-MOVEP BOARD PLAYER (CAR LST))) (END 'WIN (OTHER-PLAYER PLAYER) (CDR LST))) ((WINP PLAYER (CAR LST)) (END 'WIN PLAYER (CDR LST))) ((ENDP (CAR LST)) (END 'DRAW PLAYER (CDR LST))) (T (TIC-TAC-TOEP1 (CAR LST) (OTHER-PLAYER PLAYER) (CDR LST))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to establish that the measure (COUNT LST) decreases according to the well-founded relation LESSP in each recursive call. Hence, TIC-TAC-TOEP1 is accepted under the principle of definition. Observe that: (OR (FALSEP (TIC-TAC-TOEP1 BOARD PLAYER LST)) (LITATOM (TIC-TAC-TOEP1 BOARD PLAYER LST))) is a theorem. [ 0.0 0.0 0.0 ] TIC-TAC-TOEP1 (DEFN TIC-TAC-TOEP (LST) (IF (AND (LISTP LST) (EQUAL (CAR LST) (LIST F F F F F F F F F))) (TIC-TAC-TOEP1 (CAR LST) 'X (CDR LST)) F)) From the definition we can conclude that: (OR (FALSEP (TIC-TAC-TOEP LST)) (LITATOM (TIC-TAC-TOEP LST))) is a theorem. [ 0.0 0.0 0.0 ] TIC-TAC-TOEP (DEFN MOVE (X I BOARD) (IF (NLISTP BOARD) BOARD (IF (ZEROP I) (CONS X (CDR BOARD)) (CONS (CAR BOARD) (MOVE X (SUB1 I) (CDR BOARD)))))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each recursive call. Hence, MOVE is accepted under the principle of definition. The definition of MOVE can be justified in another way. Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definitions of ZEROP and NLISTP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Observe that (OR (LISTP (MOVE X I BOARD)) (EQUAL (MOVE X I BOARD) BOARD)) is a theorem. [ 0.0 0.0 0.0 ] MOVE (DEFN LEGALP (I BOARD) (AND (LESSP I '9) (EQUAL (NTH I BOARD) F))) Note that (OR (FALSEP (LEGALP I BOARD)) (TRUEP (LEGALP I BOARD))) is a theorem. [ 0.0 0.0 0.0 ] LEGALP (DEFN OPEN-SQUARES (BOARD) (IF (NLISTP BOARD) 0 (IF (CAR BOARD) (OPEN-SQUARES (CDR BOARD)) (ADD1 (OPEN-SQUARES (CDR BOARD)))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Hence, OPEN-SQUARES is accepted under the principle of definition. From the definition we can conclude that (NUMBERP (OPEN-SQUARES BOARD)) is a theorem. [ 0.0 0.0 0.0 ] OPEN-SQUARES (PROVE-LEMMA LEGAL-MOVES-REDUCE-OPEN-SQUARES (REWRITE) (IMPLIES (AND X (LEGALP I BOARD)) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD)))) WARNING: Note that the proposed lemma LEGAL-MOVES-REDUCE-OPEN-SQUARES is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This conjecture can be simplified, using the abbreviations LEGALP, AND, and IMPLIES, to: (IMPLIES (AND X (LESSP I 9) (EQUAL (NTH I BOARD) F)) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD))). This simplifies, clearly, to: (IMPLIES (AND X (LESSP I 9) (NOT (NTH I BOARD))) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD))), which we will name *1. We will appeal to induction. There are five plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL 9 0) (NOT (NUMBERP 9))) (p X I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p X I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p X (SUB1 I) (CDR BOARD))) (p X I BOARD))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT inform us that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme produces the following five new goals: Case 5. (IMPLIES (AND (OR (EQUAL 9 0) (NOT (NUMBERP 9))) X (LESSP I 9) (NOT (NTH I BOARD))) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD))). This simplifies, expanding the definitions of EQUAL, NUMBERP, NOT, and OR, to: T. Case 4. (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (OR (EQUAL I 0) (NOT (NUMBERP I))) X (LESSP I 9) (NOT (NTH I BOARD))) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD))). This simplifies, opening up the definitions of EQUAL, NUMBERP, NOT, OR, LESSP, NTH, MOVE, and OPEN-SQUARES, to the following four new formulas: Case 4.4. (IMPLIES (AND (EQUAL I 0) X (NOT (CAR BOARD)) (NOT (LISTP BOARD))) (LESSP (OPEN-SQUARES BOARD) 0)). But this again simplifies, rewriting with the lemma CAR-NLISTP, to: T. Case 4.3. (IMPLIES (AND (EQUAL I 0) X (NOT (CAR BOARD)) (LISTP BOARD)) (LESSP (OPEN-SQUARES (CONS X (CDR BOARD))) (ADD1 (OPEN-SQUARES (CDR BOARD))))), which again simplifies, rewriting with CDR-CONS, CAR-CONS, and SUB1-ADD1, and opening up the definitions of OPEN-SQUARES and LESSP, to the new goal: (IMPLIES (AND X (NOT (CAR BOARD)) (LISTP BOARD) (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))) (LESSP (SUB1 (OPEN-SQUARES (CDR BOARD))) (OPEN-SQUARES (CDR BOARD)))), which again simplifies, using linear arithmetic, to: T. Case 4.2. (IMPLIES (AND (NOT (NUMBERP I)) X (NOT (CAR BOARD)) (NOT (LISTP BOARD))) (LESSP (OPEN-SQUARES BOARD) 0)), which again simplifies, applying CAR-NLISTP, to: T. Case 4.1. (IMPLIES (AND (NOT (NUMBERP I)) X (NOT (CAR BOARD)) (LISTP BOARD)) (LESSP (OPEN-SQUARES (CONS X (CDR BOARD))) (ADD1 (OPEN-SQUARES (CDR BOARD))))). This again simplifies, applying CDR-CONS, CAR-CONS, and SUB1-ADD1, and unfolding the definitions of OPEN-SQUARES and LESSP, to the new goal: (IMPLIES (AND (NOT (NUMBERP I)) X (NOT (CAR BOARD)) (LISTP BOARD) (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))) (LESSP (SUB1 (OPEN-SQUARES (CDR BOARD))) (OPEN-SQUARES (CDR BOARD)))), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (LESSP (SUB1 I) 9)) X (LESSP I 9) (NOT (NTH I BOARD))) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD))), which simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP I 1) (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (LESSP (SUB1 I) 9)) X (LESSP I 9) (NOT (NTH I BOARD))) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD))). But this again simplifies, unfolding the functions SUB1, NUMBERP, EQUAL, LESSP, NOT, and OR, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NTH (SUB1 I) (CDR BOARD)) X (LESSP I 9) (NOT (NTH I BOARD))) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD))), which simplifies, expanding the definitions of EQUAL, NUMBERP, NOT, OR, LESSP, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (LESSP (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (OPEN-SQUARES (CDR BOARD))) X (LESSP I 9) (NOT (NTH I BOARD))) (LESSP (OPEN-SQUARES (MOVE X I BOARD)) (OPEN-SQUARES BOARD))), which simplifies, unfolding EQUAL, NUMBERP, NOT, OR, LESSP, SUB1, NTH, MOVE, and OPEN-SQUARES, to three new conjectures: Case 1.3. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (OPEN-SQUARES (CDR BOARD))) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD))) (NOT (LISTP BOARD))) (LESSP (OPEN-SQUARES BOARD) 0)), which again simplifies, unfolding OPEN-SQUARES and LESSP, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (OPEN-SQUARES (CDR BOARD))) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD)))) (LISTP BOARD)). But this further simplifies, rewriting with CDR-NLISTP, and unfolding the definitions of LISTP, MOVE, OPEN-SQUARES, and LESSP, to: T. Case 1.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (OPEN-SQUARES (CDR BOARD))) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD))) (LISTP BOARD) (CAR BOARD)) (LESSP (OPEN-SQUARES (CONS (CAR BOARD) (MOVE X (SUB1 I) (CDR BOARD)))) (OPEN-SQUARES (CDR BOARD)))). This again simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the function OPEN-SQUARES, to: T. Case 1.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (OPEN-SQUARES (CDR BOARD))) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD))) (LISTP BOARD) (NOT (CAR BOARD))) (LESSP (OPEN-SQUARES (CONS (CAR BOARD) (MOVE X (SUB1 I) (CDR BOARD)))) (ADD1 (OPEN-SQUARES (CDR BOARD))))). However this again simplifies, applying CDR-CONS, CAR-CONS, and SUB1-ADD1, and unfolding the functions OPEN-SQUARES and LESSP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LEGAL-MOVES-REDUCE-OPEN-SQUARES (PROVE-LEMMA MOVE-NEVER-INCREASES-OPEN-SQUARES (REWRITE) (IMPLIES X (NOT (LESSP (OPEN-SQUARES BOARD) (OPEN-SQUARES (MOVE X I BOARD)))))) WARNING: Note that the proposed lemma MOVE-NEVER-INCREASES-OPEN-SQUARES is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. Name the conjecture *1. Let us appeal to the induction principle. The recursive terms in the conjecture suggest three inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD X I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (p (CDR BOARD) X (SUB1 I))) (p BOARD X I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (p (CDR BOARD) X (SUB1 I))) (p BOARD X I))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to show that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instances chosen for I. The above induction scheme leads to the following three new goals: Case 3. (IMPLIES (AND (NLISTP BOARD) X) (NOT (LESSP (OPEN-SQUARES BOARD) (OPEN-SQUARES (MOVE X I BOARD))))). This simplifies, expanding the definitions of NLISTP, OPEN-SQUARES, MOVE, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X) (NOT (LESSP (OPEN-SQUARES BOARD) (OPEN-SQUARES (MOVE X I BOARD))))). This simplifies, unfolding NLISTP, OPEN-SQUARES, and MOVE, to the following three new goals: Case 2.3. (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X (NOT (EQUAL I 0)) (NUMBERP I)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (CONS (CAR BOARD) (MOVE X (SUB1 I) (CDR BOARD))))))). This again simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the function OPEN-SQUARES, to: T. Case 2.2. (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X (EQUAL I 0)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (CONS X (CDR BOARD)))))), which again simplifies, applying CDR-CONS and CAR-CONS, and unfolding the functions SUB1, EQUAL, MOVE, and OPEN-SQUARES, to: (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (LISTP (CDR BOARD)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (CONS X (CDDR BOARD))))) X) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (CDR BOARD))))), which again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X (NOT (NUMBERP I))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (CONS X (CDR BOARD)))))), which again simplifies, applying CDR-CONS and CAR-CONS, and unfolding OPEN-SQUARES, to: (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X (NOT (NUMBERP I))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (CDR BOARD))))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X) (NOT (LESSP (OPEN-SQUARES BOARD) (OPEN-SQUARES (MOVE X I BOARD))))), which simplifies, applying SUB1-ADD1, and expanding the functions NLISTP, OPEN-SQUARES, MOVE, and LESSP, to the following three new goals: Case 1.3. (IMPLIES (AND (LISTP BOARD) (NOT (CAR BOARD)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X (NOT (NUMBERP I)) (NOT (EQUAL (OPEN-SQUARES (CONS X (CDR BOARD))) 0))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (SUB1 (OPEN-SQUARES (CONS X (CDR BOARD))))))). However this again simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up OPEN-SQUARES, to the new conjecture: (IMPLIES (AND (LISTP BOARD) (NOT (CAR BOARD)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X (NOT (NUMBERP I)) (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (SUB1 (OPEN-SQUARES (CDR BOARD)))))), which again simplifies, using linear arithmetic, to: T. Case 1.2. (IMPLIES (AND (LISTP BOARD) (NOT (CAR BOARD)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X (EQUAL I 0) (NOT (EQUAL (OPEN-SQUARES (CONS X (CDR BOARD))) 0))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (SUB1 (OPEN-SQUARES (CONS X (CDR BOARD))))))), which again simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding the definitions of SUB1, EQUAL, MOVE, and OPEN-SQUARES, to two new formulas: Case 1.2.2. (IMPLIES (AND (LISTP BOARD) (NOT (CAR BOARD)) (NOT (LISTP (CDR BOARD))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (CDR BOARD)))) X (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (SUB1 (OPEN-SQUARES (CDR BOARD)))))), which again simplifies, using linear arithmetic, to: T. Case 1.2.1. (IMPLIES (AND (LISTP BOARD) (NOT (CAR BOARD)) (LISTP (CDR BOARD)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (CONS X (CDDR BOARD))))) X (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (SUB1 (OPEN-SQUARES (CDR BOARD)))))), which again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (LISTP BOARD) (NOT (CAR BOARD)) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))))) X (NOT (EQUAL I 0)) (NUMBERP I) (NOT (EQUAL (OPEN-SQUARES (CONS F (MOVE X (SUB1 I) (CDR BOARD)))) 0))) (NOT (LESSP (OPEN-SQUARES (CDR BOARD)) (SUB1 (OPEN-SQUARES (CONS F (MOVE X (SUB1 I) (CDR BOARD)))))))), which again simplifies, rewriting with CDR-CONS, CAR-CONS, and SUB1-ADD1, and opening up the definition of OPEN-SQUARES, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.0 ] MOVE-NEVER-INCREASES-OPEN-SQUARES (DEFN PICK-MOVE1 (BOARD I) (COND ((NLISTP BOARD) 0) ((EQUAL (CAR BOARD) F) I) (T (PICK-MOVE1 (CDR BOARD) (ADD1 I))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Hence, PICK-MOVE1 is accepted under the definitional principle. From the definition we can conclude that: (OR (NUMBERP (PICK-MOVE1 BOARD I)) (EQUAL (PICK-MOVE1 BOARD I) I)) is a theorem. [ 0.0 0.0 0.0 ] PICK-MOVE1 (DEFN PICK-MOVE (BOARD) (LET ((BOOK-MOVE (ASSOC BOARD '(((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X O X X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE O X X) . 2) ((*1*FALSE X X *1*FALSE O O O X X) . 3) ((X *1*FALSE X *1*FALSE O O O X X) . 3) ((*1*FALSE *1*FALSE X *1*FALSE O *1*FALSE O X X) . 5) ((*1*FALSE X *1*FALSE *1*FALSE O *1*FALSE O X X) . 2) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE O X X) . 2) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O *1*FALSE *1*FALSE X X) . 6) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X X O X) . 1) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE X O X) . 1) ((*1*FALSE *1*FALSE X *1*FALSE O *1*FALSE X O X) . 1) ((O X X *1*FALSE O *1*FALSE X O X) . 5) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((*1*FALSE *1*FALSE O *1*FALSE O X *1*FALSE X X) . 6) ((*1*FALSE *1*FALSE O X O X X O X) . 1) ((*1*FALSE *1*FALSE O *1*FALSE O X X *1*FALSE X) . 7) ((*1*FALSE *1*FALSE O X O X *1*FALSE *1*FALSE X) . 6) ((*1*FALSE X O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((X *1*FALSE O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X *1*FALSE *1*FALSE X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE O X X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O X O *1*FALSE X) . 2) ((*1*FALSE *1*FALSE X X O *1*FALSE O *1*FALSE X) . 5) ((*1*FALSE X *1*FALSE X O *1*FALSE O *1*FALSE X) . 2) ((X *1*FALSE *1*FALSE X O *1*FALSE O *1*FALSE X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE *1*FALSE *1*FALSE X) . 6) ((*1*FALSE *1*FALSE X *1*FALSE O O *1*FALSE X X) . 3) ((*1*FALSE *1*FALSE X *1*FALSE O O X *1*FALSE X) . 3) ((O *1*FALSE X X O O *1*FALSE X X) . 6) ((O *1*FALSE X X O O X *1*FALSE X) . 7) ((*1*FALSE X X *1*FALSE O O *1*FALSE *1*FALSE X) . 3) ((X *1*FALSE X *1*FALSE O O *1*FALSE *1*FALSE X) . 3) ((*1*FALSE *1*FALSE X *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 5) ((*1*FALSE X O *1*FALSE O *1*FALSE *1*FALSE X X) . 6) ((*1*FALSE X O *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((*1*FALSE X O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((*1*FALSE X O X O *1*FALSE *1*FALSE *1*FALSE X) . 6) ((X X O *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 6) ((*1*FALSE X *1*FALSE *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 2) ((X O X *1*FALSE O *1*FALSE O X X) . 5) ((X O *1*FALSE *1*FALSE O *1*FALSE *1*FALSE X X) . 6) ((X O *1*FALSE *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((X O *1*FALSE *1*FALSE O X *1*FALSE *1*FALSE X) . 7) ((X O *1*FALSE X O *1*FALSE *1*FALSE *1*FALSE X) . 7) ((X O X *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 7) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE X) . 4) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X O X X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE O X X) . 2) ((*1*FALSE X X *1*FALSE O O O X X) . 3) ((X *1*FALSE X *1*FALSE O O O X X) . 3) ((*1*FALSE *1*FALSE X *1*FALSE O *1*FALSE O X X) . 5) ((*1*FALSE X *1*FALSE *1*FALSE O *1*FALSE O X X) . 2) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE O X X) . 2) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O *1*FALSE *1*FALSE X X) . 6) ((X *1*FALSE X O O *1*FALSE X X O) . 5) ((X X *1*FALSE O O *1*FALSE X X O) . 5) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE X X O) . 3) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O *1*FALSE X X *1*FALSE) . 8) ((X O *1*FALSE *1*FALSE O X X X O) . 3) ((X O *1*FALSE X O X *1*FALSE X O) . 6) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X *1*FALSE X *1*FALSE) . 8) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE O X X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O X O X *1*FALSE) . 2) ((O *1*FALSE X X O *1*FALSE O X X) . 5) ((O *1*FALSE X X O X O X *1*FALSE) . 8) ((O X X X O *1*FALSE O X *1*FALSE) . 8) ((*1*FALSE X *1*FALSE X O *1*FALSE O X *1*FALSE) . 2) ((X *1*FALSE *1*FALSE X O *1*FALSE O X *1*FALSE) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((X O X X O *1*FALSE *1*FALSE X O) . 6) ((*1*FALSE *1*FALSE X *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 8) ((*1*FALSE X O *1*FALSE O *1*FALSE *1*FALSE X X) . 6) ((*1*FALSE X O X O *1*FALSE X X O) . 5) ((X X O *1*FALSE O *1*FALSE X X O) . 5) ((*1*FALSE X O *1*FALSE O *1*FALSE X X *1*FALSE) . 8) ((*1*FALSE X O *1*FALSE O X *1*FALSE X *1*FALSE) . 6) ((*1*FALSE X O X O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((X X O *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((*1*FALSE X *1*FALSE *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 2) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE O X X) . 2) ((X *1*FALSE *1*FALSE *1*FALSE O X O X *1*FALSE) . 2) ((X *1*FALSE *1*FALSE X O *1*FALSE O X *1*FALSE) . 2) ((X O X *1*FALSE O *1*FALSE O X X) . 5) ((X O X *1*FALSE O X O X *1*FALSE) . 8) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE X *1*FALSE) . 4) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X X O X) . 1) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE X O X) . 1) ((*1*FALSE *1*FALSE X *1*FALSE O *1*FALSE X O X) . 1) ((O X X *1*FALSE O *1*FALSE X O X) . 5) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((X *1*FALSE X O O *1*FALSE X X O) . 5) ((X X *1*FALSE O O *1*FALSE X X O) . 5) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE X X O) . 3) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O *1*FALSE X X *1*FALSE) . 8) ((X *1*FALSE *1*FALSE *1*FALSE O X X *1*FALSE O) . 3) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE *1*FALSE X O *1*FALSE X *1*FALSE X) . 7) ((O *1*FALSE *1*FALSE X O *1*FALSE X X *1*FALSE) . 8) ((O *1*FALSE *1*FALSE X O X X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE X X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O X *1*FALSE X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((*1*FALSE O X *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((*1*FALSE O X *1*FALSE O *1*FALSE X X *1*FALSE) . 8) ((*1*FALSE O X *1*FALSE O X X *1*FALSE *1*FALSE) . 7) ((*1*FALSE O X X O *1*FALSE X *1*FALSE *1*FALSE) . 7) ((X O X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 7) ((*1*FALSE *1*FALSE X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 1) ((O X X *1*FALSE O *1*FALSE X O X) . 5) ((O X *1*FALSE *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((O X *1*FALSE *1*FALSE O *1*FALSE X X *1*FALSE) . 8) ((O X *1*FALSE *1*FALSE O X X *1*FALSE *1*FALSE) . 8) ((O X *1*FALSE X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O X X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((X *1*FALSE *1*FALSE O O *1*FALSE X *1*FALSE X) . 5) ((X *1*FALSE *1*FALSE O O *1*FALSE X X *1*FALSE) . 5) ((X O *1*FALSE O O X X *1*FALSE X) . 7) ((X O *1*FALSE O O X X X *1*FALSE) . 8) ((X *1*FALSE X O O *1*FALSE X *1*FALSE *1*FALSE) . 5) ((X X *1*FALSE O O *1*FALSE X *1*FALSE *1*FALSE) . 5) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 3) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE X *1*FALSE *1*FALSE) . 4) ((*1*FALSE *1*FALSE O *1*FALSE O X *1*FALSE X X) . 6) ((*1*FALSE *1*FALSE O X O X X O X) . 1) ((*1*FALSE *1*FALSE O *1*FALSE O X X *1*FALSE X) . 7) ((*1*FALSE *1*FALSE O X O X *1*FALSE *1*FALSE X) . 6) ((*1*FALSE X O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((X *1*FALSE O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X *1*FALSE *1*FALSE X) . 2) ((X O *1*FALSE *1*FALSE O X X X O) . 3) ((X O *1*FALSE X O X *1*FALSE X O) . 6) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X *1*FALSE X *1*FALSE) . 8) ((X *1*FALSE *1*FALSE *1*FALSE O X X *1*FALSE O) . 3) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE O X X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE *1*FALSE X O X *1*FALSE *1*FALSE X) . 2) ((O *1*FALSE *1*FALSE X O X *1*FALSE X *1*FALSE) . 8) ((O *1*FALSE *1*FALSE X O X X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE X X O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X *1*FALSE X O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((X O X *1*FALSE O X X *1*FALSE O) . 7) ((X O X X O X *1*FALSE *1*FALSE O) . 7) ((*1*FALSE *1*FALSE X *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((*1*FALSE X O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((*1*FALSE X O *1*FALSE O X *1*FALSE X *1*FALSE) . 6) ((O X O *1*FALSE O X X *1*FALSE X) . 7) ((O X O *1*FALSE O X X X *1*FALSE) . 8) ((O X O X O X X *1*FALSE *1*FALSE) . 8) ((*1*FALSE X O X O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((X X O *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((*1*FALSE X *1*FALSE *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 2) ((X *1*FALSE O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((X *1*FALSE O *1*FALSE O X *1*FALSE X *1*FALSE) . 6) ((X *1*FALSE O O O X X *1*FALSE X) . 7) ((X *1*FALSE O O O X X X *1*FALSE) . 8) ((X *1*FALSE O *1*FALSE O X X *1*FALSE *1*FALSE) . 3) ((X *1*FALSE O X O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((X X O *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((X *1*FALSE *1*FALSE *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 2) ((*1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE X *1*FALSE *1*FALSE *1*FALSE) . 4) ((O X O *1*FALSE X *1*FALSE *1*FALSE *1*FALSE X) . 7) ((O *1*FALSE *1*FALSE *1*FALSE X *1*FALSE *1*FALSE *1*FALSE X) . 2) ((O O X *1*FALSE X *1*FALSE *1*FALSE X *1*FALSE) . 6) ((O X O *1*FALSE X *1*FALSE X *1*FALSE *1*FALSE) . 7) ((O *1*FALSE *1*FALSE *1*FALSE X *1*FALSE X *1*FALSE *1*FALSE) . 2) ((O *1*FALSE *1*FALSE O X X *1*FALSE *1*FALSE X) . 6) ((O *1*FALSE *1*FALSE O X X 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((O X X X X *1*FALSE O O *1*FALSE) . 8) ((O X X *1*FALSE X *1*FALSE *1*FALSE O *1*FALSE) . 6) ((O X *1*FALSE *1*FALSE X *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 7) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE O X X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O X O *1*FALSE X) . 2) ((*1*FALSE *1*FALSE X X O *1*FALSE O *1*FALSE X) . 5) ((*1*FALSE X *1*FALSE X O *1*FALSE O *1*FALSE X) . 2) ((X *1*FALSE *1*FALSE X O *1*FALSE O *1*FALSE X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE *1*FALSE *1*FALSE X) . 6) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE O X X) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O X O X *1*FALSE) . 2) ((O *1*FALSE X X O *1*FALSE O X X) . 5) ((O *1*FALSE X X O X O X *1*FALSE) . 8) ((O X X X O *1*FALSE O X *1*FALSE) . 8) ((*1*FALSE X *1*FALSE X O *1*FALSE O X *1*FALSE) . 2) ((X *1*FALSE *1*FALSE X O *1*FALSE O X *1*FALSE) . 2) ((*1*FALSE *1*FALSE *1*FALSE X O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((O *1*FALSE *1*FALSE X O *1*FALSE X *1*FALSE X) . 7) ((O *1*FALSE *1*FALSE X O *1*FALSE X X *1*FALSE) . 8) ((O *1*FALSE *1*FALSE X O X X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE X X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O X *1*FALSE X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE *1*FALSE X O X *1*FALSE *1*FALSE X) . 2) ((O *1*FALSE *1*FALSE X O X *1*FALSE X *1*FALSE) . 8) ((O *1*FALSE *1*FALSE X O X X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE X X O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X *1*FALSE X O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O *1*FALSE X X O O *1*FALSE X X) . 6) ((O *1*FALSE X X O O X *1*FALSE X) . 7) ((O *1*FALSE X X O *1*FALSE *1*FALSE *1*FALSE X) . 5) ((O *1*FALSE X X O *1*FALSE *1*FALSE X *1*FALSE) . 8) ((O *1*FALSE X X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE X X O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X X X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X O X O *1*FALSE *1*FALSE X X) . 6) ((O X O X O *1*FALSE X *1*FALSE X) . 7) ((O X *1*FALSE X O *1*FALSE *1*FALSE X *1*FALSE) . 8) ((O X *1*FALSE X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O X *1*FALSE 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*1*FALSE X X *1*FALSE) . 8) ((*1*FALSE O X *1*FALSE O X X *1*FALSE *1*FALSE) . 7) ((*1*FALSE O X X O *1*FALSE X *1*FALSE *1*FALSE) . 7) ((X O X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 7) ((*1*FALSE *1*FALSE X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 1) ((X O X *1*FALSE O X X *1*FALSE O) . 7) ((X O X X O X *1*FALSE *1*FALSE O) . 7) ((*1*FALSE *1*FALSE X *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O *1*FALSE X X O O *1*FALSE X X) . 6) ((O *1*FALSE X X O O X *1*FALSE X) . 7) ((O *1*FALSE X X O *1*FALSE *1*FALSE *1*FALSE X) . 5) ((O *1*FALSE X X O *1*FALSE *1*FALSE X *1*FALSE) . 8) ((O *1*FALSE X X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O *1*FALSE X X O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X X X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X X *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 5) ((O X X *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 8) ((O X X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O X X *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X X X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 8) ((X O X *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 7) ((X O X O O X *1*FALSE X *1*FALSE) . 8) ((X O X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 7) ((X O X *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 7) ((X O X X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 7) ((*1*FALSE *1*FALSE X *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 4) ((*1*FALSE X O *1*FALSE O *1*FALSE *1*FALSE X X) . 6) ((*1*FALSE X O *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((*1*FALSE X O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((*1*FALSE X O X O *1*FALSE *1*FALSE *1*FALSE X) . 6) ((X X O *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 6) ((*1*FALSE X *1*FALSE *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 2) ((*1*FALSE X O *1*FALSE O *1*FALSE *1*FALSE X X) . 6) ((*1*FALSE X O X O *1*FALSE X X O) . 5) ((X X O *1*FALSE O *1*FALSE X X O) . 5) ((*1*FALSE X O *1*FALSE O *1*FALSE X X *1*FALSE) . 8) ((*1*FALSE X O *1*FALSE O X *1*FALSE X *1*FALSE) . 6) ((*1*FALSE X O X O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((X X O *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((*1*FALSE X *1*FALSE *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 2) ((O X X *1*FALSE O *1*FALSE X O X) . 5) ((O X *1*FALSE *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((O X *1*FALSE *1*FALSE O *1*FALSE X X *1*FALSE) . 8) ((O X *1*FALSE *1*FALSE O X X *1*FALSE *1*FALSE) . 8) ((O X *1*FALSE X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O X X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((*1*FALSE X O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((*1*FALSE X O *1*FALSE O X *1*FALSE X *1*FALSE) . 6) ((O X O *1*FALSE O X X *1*FALSE X) . 7) ((O X O *1*FALSE O X X X *1*FALSE) . 8) ((O X O X O X X *1*FALSE *1*FALSE) . 8) ((*1*FALSE X O X O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((X X O *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((*1*FALSE X *1*FALSE *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 2) ((O X O X O *1*FALSE *1*FALSE X X) . 6) ((O X O X O *1*FALSE X *1*FALSE X) . 7) ((O X *1*FALSE X O *1*FALSE *1*FALSE X *1*FALSE) . 8) ((O X *1*FALSE X O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O X *1*FALSE X O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X X X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X X *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 5) ((O X X *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 8) ((O X X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 8) ((O X X *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 8) ((O X X X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 8) ((X X O *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 6) ((X X O *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((X X O *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((X X O X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 6) ((*1*FALSE X *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 4) ((X O X *1*FALSE O *1*FALSE O X X) . 5) ((X O *1*FALSE *1*FALSE O *1*FALSE *1*FALSE X X) . 6) ((X O *1*FALSE *1*FALSE O *1*FALSE X *1*FALSE X) . 7) ((X O *1*FALSE *1*FALSE O X *1*FALSE *1*FALSE X) . 7) ((X O *1*FALSE X O *1*FALSE *1*FALSE *1*FALSE X) . 7) ((X O X *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 7) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE O X X) . 2) ((X *1*FALSE *1*FALSE *1*FALSE O X O X *1*FALSE) . 2) ((X *1*FALSE *1*FALSE X O *1*FALSE O X *1*FALSE) . 2) ((X O X *1*FALSE O *1*FALSE O X X) . 5) ((X O X *1*FALSE O X O X *1*FALSE) . 8) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((X *1*FALSE *1*FALSE O O *1*FALSE X *1*FALSE X) . 5) ((X *1*FALSE *1*FALSE O O *1*FALSE X X *1*FALSE) . 5) ((X O *1*FALSE O O X X *1*FALSE X) . 7) ((X O *1*FALSE O O X X X *1*FALSE) . 8) ((X *1*FALSE X O O *1*FALSE X *1*FALSE *1*FALSE) . 5) ((X X *1*FALSE O O *1*FALSE X *1*FALSE *1*FALSE) . 5) ((X *1*FALSE *1*FALSE *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 3) ((X *1*FALSE O *1*FALSE O X *1*FALSE *1*FALSE X) . 6) ((X *1*FALSE O *1*FALSE O X *1*FALSE X *1*FALSE) . 6) ((X *1*FALSE O O O X X *1*FALSE X) . 7) ((X *1*FALSE O O O X X X *1*FALSE) . 8) ((X *1*FALSE O *1*FALSE O X X *1*FALSE *1*FALSE) . 3) ((X *1*FALSE O X O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((X X O *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((X *1*FALSE *1*FALSE *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 2) ((X *1*FALSE *1*FALSE X O *1*FALSE O *1*FALSE X) . 2) ((X *1*FALSE *1*FALSE X O *1*FALSE O X *1*FALSE) . 2) ((X *1*FALSE *1*FALSE X O X O *1*FALSE *1*FALSE) . 2) ((X O X X O *1*FALSE O *1*FALSE X) . 7) ((X *1*FALSE *1*FALSE X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 6) ((X O X *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 7) ((X O X O O X *1*FALSE X *1*FALSE) . 8) ((X O X *1*FALSE O *1*FALSE X *1*FALSE *1*FALSE) . 7) ((X O X *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 7) ((X O X X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 7) ((X X O *1*FALSE O *1*FALSE *1*FALSE *1*FALSE X) . 6) ((X X O *1*FALSE O *1*FALSE *1*FALSE X *1*FALSE) . 6) ((X X O *1*FALSE O X *1*FALSE *1*FALSE *1*FALSE) . 6) ((X X O X O *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 6) ((X *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE *1*FALSE) . 4))))) (IF BOOK-MOVE (CDR BOOK-MOVE) (PICK-MOVE1 BOARD 0)))) [ 0.1 0.0 0.0 ] PICK-MOVE (DISABLE PICK-MOVE) [ 0.0 0.0 0.0 ] PICK-MOVE-OFF (DEFN TIC-TAC-TOE1 (XMOVES BOARD) (LET ((BOARD1 (MOVE 'X (CAR XMOVES) BOARD))) (IF (LEGALP (CAR XMOVES) BOARD) (IF (ENDP BOARD1) (LIST BOARD1 'DRAW) (LET ((BOARD2 (MOVE 'O (PICK-MOVE BOARD1) BOARD1))) (IF (WINP 'O BOARD2) (LIST BOARD1 BOARD2 'O-WIN) (CONS BOARD1 (CONS BOARD2 (TIC-TAC-TOE1 (CDR XMOVES) BOARD2)))))) (LIST BOARD1 'O-WIN))) ((LESSP (OPEN-SQUARES BOARD)))) Linear arithmetic and the lemmas MOVE-NEVER-INCREASES-OPEN-SQUARES and LEGAL-MOVES-REDUCE-OPEN-SQUARES inform us that the measure (OPEN-SQUARES BOARD) decreases according to the well-founded relation LESSP in each recursive call. Hence, TIC-TAC-TOE1 is accepted under the principle of definition. From the definition we can conclude that: (LISTP (TIC-TAC-TOE1 XMOVES BOARD)) is a theorem. [ 0.1 0.0 0.0 ] TIC-TAC-TOE1 (DEFN TIC-TAC-TOE (XMOVES) (LET ((BOARD0 (LIST F F F F F F F F F))) (CONS BOARD0 (TIC-TAC-TOE1 XMOVES BOARD0)))) Note that (LISTP (TIC-TAC-TOE XMOVES)) is a theorem. [ 0.0 0.0 0.0 ] TIC-TAC-TOE (DEFN LENGTH (X) (IF (NLISTP X) 0 (ADD1 (LENGTH (CDR X))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT X) decreases according to the well-founded relation LESSP in each recursive call. Hence, LENGTH is accepted under the principle of definition. From the definition we can conclude that (NUMBERP (LENGTH X)) is a theorem. [ 0.0 0.0 0.0 ] LENGTH (PROVE-LEMMA LENGTH-MOVE (REWRITE) (EQUAL (LENGTH (MOVE X I BOARD)) (LENGTH BOARD))) Name the conjecture *1. Perhaps we can prove it by induction. There are three plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p X I BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I)) (p X I BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (p X (SUB1 I) (CDR BOARD))) (p X I BOARD))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme produces the following three new goals: Case 3. (IMPLIES (NLISTP BOARD) (EQUAL (LENGTH (MOVE X I BOARD)) (LENGTH BOARD))). This simplifies, expanding NLISTP, MOVE, LENGTH, and EQUAL, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I)) (EQUAL (LENGTH (MOVE X I BOARD)) (LENGTH BOARD))). This simplifies, rewriting with CDR-CONS, and unfolding the definitions of NLISTP, ZEROP, EQUAL, MOVE, and LENGTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (EQUAL (LENGTH (MOVE X (SUB1 I) (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (LENGTH (MOVE X I BOARD)) (LENGTH BOARD))), which simplifies, rewriting with CDR-CONS, and expanding the definitions of NLISTP, ZEROP, MOVE, and LENGTH, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LENGTH-MOVE (DEFN LEGAL-BOOKP (ALIST) (IF (NLISTP ALIST) T (AND (LEGALP (CDAR ALIST) (CAAR ALIST)) (LEGAL-BOOKP (CDR ALIST))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to show that the measure (COUNT ALIST) decreases according to the well-founded relation LESSP in each recursive call. Hence, LEGAL-BOOKP is accepted under the definitional principle. Observe that: (OR (FALSEP (LEGAL-BOOKP ALIST)) (TRUEP (LEGAL-BOOKP ALIST))) is a theorem. [ 0.0 0.0 0.0 ] LEGAL-BOOKP (PROVE-LEMMA LEGAL-BOOKP-IMPLIES-LEGALP-CDR-ASSOC (REWRITE) (IMPLIES (AND (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (LEGALP (CDR (ASSOC BOARD ALIST)) BOARD))) WARNING: Note that the rewrite rule LEGAL-BOOKP-IMPLIES-LEGALP-CDR-ASSOC will be stored so as to apply only to terms with the nonrecursive function symbol LEGALP. This conjecture simplifies, expanding the definition of LEGALP, to the following two new formulas: Case 2. (IMPLIES (AND (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (LESSP (CDR (ASSOC BOARD ALIST)) 9)). Name the above subgoal *1. Case 1. (IMPLIES (AND (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (NOT (NTH (CDR (ASSOC BOARD ALIST)) BOARD))), which we would usually push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us consider: (IMPLIES (AND (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (LEGALP (CDR (ASSOC BOARD ALIST)) BOARD)), which we named *1 above. We will appeal to induction. Three inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP ALIST) (p BOARD ALIST)) (IMPLIES (AND (NOT (NLISTP ALIST)) (EQUAL BOARD (CAAR ALIST))) (p BOARD ALIST)) (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (p BOARD (CDR ALIST))) (p BOARD ALIST))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to show that the measure (COUNT ALIST) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces five new conjectures: Case 5. (IMPLIES (AND (NLISTP ALIST) (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (LEGALP (CDR (ASSOC BOARD ALIST)) BOARD)), which simplifies, unfolding the definitions of NLISTP, LEGAL-BOOKP, and ASSOC, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP ALIST)) (EQUAL BOARD (CAAR ALIST)) (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (LEGALP (CDR (ASSOC BOARD ALIST)) BOARD)), which simplifies, expanding NLISTP, LEGAL-BOOKP, LEGALP, ASSOC, and EQUAL, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (NOT (LEGAL-BOOKP (CDR ALIST))) (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (LEGALP (CDR (ASSOC BOARD ALIST)) BOARD)), which simplifies, expanding the functions NLISTP, LEGAL-BOOKP, and LEGALP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (NOT (ASSOC BOARD (CDR ALIST))) (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (LEGALP (CDR (ASSOC BOARD ALIST)) BOARD)), which simplifies, opening up NLISTP, LEGAL-BOOKP, LEGALP, and ASSOC, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (LEGALP (CDR (ASSOC BOARD (CDR ALIST))) BOARD) (LEGAL-BOOKP ALIST) (ASSOC BOARD ALIST)) (LEGALP (CDR (ASSOC BOARD ALIST)) BOARD)), which simplifies, unfolding the definitions of NLISTP, LEGALP, LEGAL-BOOKP, ASSOC, and EQUAL, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LEGAL-BOOKP-IMPLIES-LEGALP-CDR-ASSOC (DEFN PICK-MOVE2 (BOARD) (COND ((NLISTP BOARD) 0) ((EQUAL (CAR BOARD) F) 0) (T (ADD1 (PICK-MOVE2 (CDR BOARD)))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Hence, PICK-MOVE2 is accepted under the definitional principle. From the definition we can conclude that (NUMBERP (PICK-MOVE2 BOARD)) is a theorem. [ 0.0 0.0 0.0 ] PICK-MOVE2 (PROVE-LEMMA NOT-ENDP-IMPLIES-LEGALP-PICK-MOVE2 (REWRITE) (IMPLIES (AND (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LEGALP (PICK-MOVE2 BOARD) BOARD))) WARNING: Note that the rewrite rule NOT-ENDP-IMPLIES-LEGALP-PICK-MOVE2 will be stored so as to apply only to terms with the nonrecursive function symbol LEGALP. This conjecture simplifies, unfolding the definition of LEGALP, to two new conjectures: Case 2. (IMPLIES (AND (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LESSP (PICK-MOVE2 BOARD) 9)), which we will name *1. Case 1. (IMPLIES (AND (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (NOT (NTH (PICK-MOVE2 BOARD) BOARD))), which we would usually push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us consider: (IMPLIES (AND (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LEGALP (PICK-MOVE2 BOARD) BOARD)), which we named *1 above. We will appeal to induction. The recursive terms in the conjecture suggest three inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (p (CDR BOARD))) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD))) (p BOARD))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to the following five new conjectures: Case 5. (IMPLIES (AND (NLISTP BOARD) (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LEGALP (PICK-MOVE2 BOARD) BOARD)). This simplifies, expanding the definitions of NLISTP and ENDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (ENDP (CDR BOARD)) (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LEGALP (PICK-MOVE2 BOARD) BOARD)). This simplifies, unfolding NLISTP and ENDP, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (NOT (LESSP (LENGTH (CDR BOARD)) 10)) (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LEGALP (PICK-MOVE2 BOARD) BOARD)). This simplifies, applying SUB1-ADD1, and unfolding the definitions of NLISTP, ENDP, LENGTH, SUB1, NUMBERP, EQUAL, LESSP, PICK-MOVE2, NTH, and LEGALP, to two new formulas: Case 3.2. (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (NOT (LESSP (LENGTH (CDR BOARD)) 10)) (NOT (ENDP (CDR BOARD))) (LESSP (LENGTH (CDR BOARD)) 9)) (LESSP (PICK-MOVE2 (CDR BOARD)) 8)), which again simplifies, using linear arithmetic, to: T. Case 3.1. (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (NOT (LESSP (LENGTH (CDR BOARD)) 10)) (NOT (ENDP (CDR BOARD))) (LESSP (LENGTH (CDR BOARD)) 9)) (NOT (NTH (PICK-MOVE2 (CDR BOARD)) (CDR BOARD)))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (LEGALP (PICK-MOVE2 (CDR BOARD)) (CDR BOARD)) (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LEGALP (PICK-MOVE2 BOARD) BOARD)), which simplifies, applying the lemma SUB1-ADD1, and unfolding the definitions of NLISTP, LEGALP, ENDP, LENGTH, SUB1, NUMBERP, EQUAL, LESSP, PICK-MOVE2, and NTH, to the formula: (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (LESSP (PICK-MOVE2 (CDR BOARD)) 9) (NOT (NTH (PICK-MOVE2 (CDR BOARD)) (CDR BOARD))) (NOT (ENDP (CDR BOARD))) (LESSP (LENGTH (CDR BOARD)) 9)) (LESSP (PICK-MOVE2 (CDR BOARD)) 8)). However this again simplifies, using linear arithmetic, to: (IMPLIES (AND (EQUAL (PICK-MOVE2 (CDR BOARD)) 8) (LISTP BOARD) (CAR BOARD) (LESSP 8 9) (NOT (NTH 8 (CDR BOARD))) (NOT (ENDP (CDR BOARD))) (LESSP (LENGTH (CDR BOARD)) 9)) (LESSP 8 8)). However this again simplifies, unfolding the function LESSP, to: (IMPLIES (AND (EQUAL (PICK-MOVE2 (CDR BOARD)) 8) (LISTP BOARD) (CAR BOARD) (NOT (NTH 8 (CDR BOARD))) (NOT (ENDP (CDR BOARD)))) (NOT (LESSP (LENGTH (CDR BOARD)) 9))). Appealing to the lemma CAR-CDR-ELIM, we now replace BOARD by (CONS Z X) to eliminate (CDR BOARD) and (CAR BOARD). The result is: (IMPLIES (AND (EQUAL (PICK-MOVE2 X) 8) Z (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))). Eliminate the irrelevant term. We would thus like to prove: (IMPLIES (AND (EQUAL (PICK-MOVE2 X) 8) (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))), which we will finally name *1.1. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LEGALP (PICK-MOVE2 BOARD) BOARD)). This simplifies, appealing to the lemma SUB1-ADD1, and expanding the definitions of NLISTP, ENDP, LENGTH, SUB1, NUMBERP, EQUAL, LESSP, PICK-MOVE2, NTH, and LEGALP, to: T. So let us turn our attention to: (IMPLIES (AND (EQUAL (PICK-MOVE2 X) 8) (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))), which is formula *1.1 above. We will appeal to induction. The recursive terms in the conjecture suggest three inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP X) (p X)) (IMPLIES (AND (NOT (NLISTP X)) (EQUAL (CAR X) F)) (p X)) (IMPLIES (AND (NOT (NLISTP X)) (NOT (EQUAL (CAR X) F)) (p (CDR X))) (p X))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to the following six new goals: Case 6. (IMPLIES (AND (NLISTP X) (EQUAL (PICK-MOVE2 X) 8) (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))). This simplifies, expanding the functions NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP X)) (EQUAL (CAR X) F) (EQUAL (PICK-MOVE2 X) 8) (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))). This simplifies, unfolding the definitions of NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP X)) (NOT (EQUAL (CAR X) F)) (NOT (EQUAL (PICK-MOVE2 (CDR X)) 8)) (EQUAL (PICK-MOVE2 X) 8) (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))). This simplifies, rewriting with the lemmas ADD1-EQUAL and SUB1-ADD1, and opening up NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP X) (CAR X) (NOT (EQUAL (PICK-MOVE2 (CDR X)) 8)) (EQUAL (PICK-MOVE2 (CDR X)) 7) (NOT (NTH 8 X)) (NOT (ENDP (CDR X)))) (NOT (LESSP (LENGTH (CDR X)) 8))), which again simplifies, opening up the definition of EQUAL, to the formula: (IMPLIES (AND (LISTP X) (CAR X) (EQUAL (PICK-MOVE2 (CDR X)) 7) (NOT (NTH 8 X)) (NOT (ENDP (CDR X)))) (NOT (LESSP (LENGTH (CDR X)) 8))). Appealing to the lemma CAR-CDR-ELIM, we now replace X by (CONS Z V) to eliminate (CAR X) and (CDR X). This generates: (IMPLIES (AND Z (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 8 (CONS Z V))) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))). But this further simplifies, rewriting with CDR-CONS, and unfolding the definitions of SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND Z (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))), which has an irrelevant term in it. By eliminating the term we get: (IMPLIES (AND (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))), which we will finally name *1.1.1. Case 3. (IMPLIES (AND (NOT (NLISTP X)) (NOT (EQUAL (CAR X) F)) (NTH 8 (CDR X)) (EQUAL (PICK-MOVE2 X) 8) (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))). This simplifies, rewriting with ADD1-EQUAL and SUB1-ADD1, and unfolding the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP X) (CAR X) (NTH 8 (CDR X)) (EQUAL (PICK-MOVE2 (CDR X)) 7) (NOT (NTH 8 X)) (NOT (ENDP (CDR X)))) (NOT (LESSP (LENGTH (CDR X)) 8))). Appealing to the lemma CAR-CDR-ELIM, we now replace X by (CONS Z V) to eliminate (CAR X) and (CDR X). We must thus prove the formula: (IMPLIES (AND Z (NTH 8 V) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 8 (CONS Z V))) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))). However this further simplifies, appealing to the lemma CDR-CONS, and expanding the definitions of SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND Z (NTH 8 V) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))). Eliminate the irrelevant term. We would thus like to prove: (IMPLIES (AND (NTH 8 V) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))), which we will name *1.1.2. Case 2. (IMPLIES (AND (NOT (NLISTP X)) (NOT (EQUAL (CAR X) F)) (ENDP (CDR X)) (EQUAL (PICK-MOVE2 X) 8) (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))). This simplifies, rewriting with ADD1-EQUAL, and expanding the definitions of NLISTP, PICK-MOVE2, NUMBERP, and ENDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP X)) (NOT (EQUAL (CAR X) F)) (NOT (LESSP (LENGTH (CDR X)) 9)) (EQUAL (PICK-MOVE2 X) 8) (NOT (NTH 8 X)) (NOT (ENDP X))) (NOT (LESSP (LENGTH X) 9))), which simplifies, rewriting with ADD1-EQUAL and SUB1-ADD1, and opening up the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP X) (CAR X) (NOT (LESSP (LENGTH (CDR X)) 9)) (EQUAL (PICK-MOVE2 (CDR X)) 7) (NOT (NTH 8 X)) (NOT (ENDP (CDR X)))) (NOT (LESSP (LENGTH (CDR X)) 8))), which again simplifies, using linear arithmetic, to: T. So we now return to: (IMPLIES (AND (NTH 8 V) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))), which we named *1.1.2 above. Ah ha! This conjecture is subsumed by the subgoal we named *1.1.1 above. So let us turn our attention to: (IMPLIES (AND (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))), which we named *1.1.1 above. Let us appeal to the induction principle. Three inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP V) (p V)) (IMPLIES (AND (NOT (NLISTP V)) (EQUAL (CAR V) F)) (p V)) (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (p (CDR V))) (p V))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT V) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces six new goals: Case 6. (IMPLIES (AND (NLISTP V) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))), which simplifies, expanding the functions NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP V)) (EQUAL (CAR V) F) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))), which simplifies, opening up the functions NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NOT (EQUAL (PICK-MOVE2 (CDR V)) 7)) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))), which simplifies, rewriting with ADD1-EQUAL and SUB1-ADD1, and opening up the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP V) (CAR V) (NOT (EQUAL (PICK-MOVE2 (CDR V)) 7)) (EQUAL (PICK-MOVE2 (CDR V)) 6) (NOT (NTH 7 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 7))), which again simplifies, expanding EQUAL, to: (IMPLIES (AND (LISTP V) (CAR V) (EQUAL (PICK-MOVE2 (CDR V)) 6) (NOT (NTH 7 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 7))). Appealing to the lemma CAR-CDR-ELIM, we now replace V by (CONS X Z) to eliminate (CAR V) and (CDR V). We must thus prove: (IMPLIES (AND X (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 7 (CONS X Z))) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))). But this further simplifies, rewriting with the lemma CDR-CONS, and unfolding the functions SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))). Eliminate the irrelevant term. This produces: (IMPLIES (AND (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))), which we will finally name *1.1.1.1. Case 3. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NTH 7 (CDR V)) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))). This simplifies, applying the lemmas ADD1-EQUAL and SUB1-ADD1, and expanding the definitions of NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to the new formula: (IMPLIES (AND (LISTP V) (CAR V) (NTH 7 (CDR V)) (EQUAL (PICK-MOVE2 (CDR V)) 6) (NOT (NTH 7 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 7))). Applying the lemma CAR-CDR-ELIM, replace V by (CONS X Z) to eliminate (CAR V) and (CDR V). We would thus like to prove: (IMPLIES (AND X (NTH 7 Z) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 7 (CONS X Z))) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))), which further simplifies, applying the lemma CDR-CONS, and opening up the definitions of SUB1, NUMBERP, EQUAL, and NTH, to the conjecture: (IMPLIES (AND X (NTH 7 Z) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))). Eliminate the irrelevant term. We thus obtain the new goal: (IMPLIES (AND (NTH 7 Z) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))), which we will name *1.1.1.2. Case 2. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (ENDP (CDR V)) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))). This simplifies, applying the lemma ADD1-EQUAL, and expanding NLISTP, PICK-MOVE2, NUMBERP, and ENDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NOT (LESSP (LENGTH (CDR V)) 8)) (EQUAL (PICK-MOVE2 V) 7) (NOT (NTH 7 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 8))). This simplifies, applying ADD1-EQUAL and SUB1-ADD1, and expanding NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to the goal: (IMPLIES (AND (LISTP V) (CAR V) (NOT (LESSP (LENGTH (CDR V)) 8)) (EQUAL (PICK-MOVE2 (CDR V)) 6) (NOT (NTH 7 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 7))). However this again simplifies, using linear arithmetic, to: T. So we now return to: (IMPLIES (AND (NTH 7 Z) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))), which is formula *1.1.1.2 above. But this conjecture is subsumed by the subgoal we named *1.1.1.1 above. So let us turn our attention to: (IMPLIES (AND (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))), named *1.1.1.1 above. Let us appeal to the induction principle. Three inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP Z) (p Z)) (IMPLIES (AND (NOT (NLISTP Z)) (EQUAL (CAR Z) F)) (p Z)) (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (p (CDR Z))) (p Z))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to prove that the measure (COUNT Z) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to the following six new goals: Case 6. (IMPLIES (AND (NLISTP Z) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))). This simplifies, unfolding the functions NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP Z)) (EQUAL (CAR Z) F) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))). This simplifies, unfolding the definitions of NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NOT (EQUAL (PICK-MOVE2 (CDR Z)) 6)) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))). This simplifies, rewriting with ADD1-EQUAL and SUB1-ADD1, and expanding NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP Z) (CAR Z) (NOT (EQUAL (PICK-MOVE2 (CDR Z)) 6)) (EQUAL (PICK-MOVE2 (CDR Z)) 5) (NOT (NTH 6 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 6))). But this again simplifies, opening up EQUAL, to: (IMPLIES (AND (LISTP Z) (CAR Z) (EQUAL (PICK-MOVE2 (CDR Z)) 5) (NOT (NTH 6 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 6))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS X V) to eliminate (CAR Z) and (CDR Z). The result is: (IMPLIES (AND X (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 6 (CONS X V))) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))). This further simplifies, applying the lemma CDR-CONS, and expanding SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))). Eliminate the irrelevant term. We thus obtain: (IMPLIES (AND (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))), which we will finally name *1.1.1.1.1. Case 3. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NTH 6 (CDR Z)) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))). This simplifies, applying ADD1-EQUAL and SUB1-ADD1, and expanding the definitions of NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP Z) (CAR Z) (NTH 6 (CDR Z)) (EQUAL (PICK-MOVE2 (CDR Z)) 5) (NOT (NTH 6 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 6))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS X V) to eliminate (CAR Z) and (CDR Z). The result is: (IMPLIES (AND X (NTH 6 V) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 6 (CONS X V))) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))). But this further simplifies, applying CDR-CONS, and opening up the functions SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (NTH 6 V) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))), which has an irrelevant term in it. By eliminating the term we get: (IMPLIES (AND (NTH 6 V) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))), which we will name *1.1.1.1.2. Case 2. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (ENDP (CDR Z)) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))). This simplifies, rewriting with ADD1-EQUAL, and unfolding NLISTP, PICK-MOVE2, NUMBERP, and ENDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NOT (LESSP (LENGTH (CDR Z)) 7)) (EQUAL (PICK-MOVE2 Z) 6) (NOT (NTH 6 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 7))), which simplifies, rewriting with the lemmas ADD1-EQUAL and SUB1-ADD1, and opening up NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP Z) (CAR Z) (NOT (LESSP (LENGTH (CDR Z)) 7)) (EQUAL (PICK-MOVE2 (CDR Z)) 5) (NOT (NTH 6 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 6))). But this again simplifies, using linear arithmetic, to: T. So next consider: (IMPLIES (AND (NTH 6 V) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))), which is formula *1.1.1.1.2 above. You probably did not notice, but this conjecture is subsumed by another subgoal awaiting our attention, namely *1.1.1.1.1 above. So next consider: (IMPLIES (AND (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))), named *1.1.1.1.1 above. Let us appeal to the induction principle. There are three plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP V) (p V)) (IMPLIES (AND (NOT (NLISTP V)) (EQUAL (CAR V) F)) (p V)) (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (p (CDR V))) (p V))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT V) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates six new formulas: Case 6. (IMPLIES (AND (NLISTP V) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))), which simplifies, opening up NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP V)) (EQUAL (CAR V) F) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))), which simplifies, unfolding the definitions of NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NOT (EQUAL (PICK-MOVE2 (CDR V)) 5)) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))), which simplifies, rewriting with ADD1-EQUAL and SUB1-ADD1, and expanding the definitions of NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP V) (CAR V) (NOT (EQUAL (PICK-MOVE2 (CDR V)) 5)) (EQUAL (PICK-MOVE2 (CDR V)) 4) (NOT (NTH 5 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 5))), which again simplifies, opening up the function EQUAL, to the formula: (IMPLIES (AND (LISTP V) (CAR V) (EQUAL (PICK-MOVE2 (CDR V)) 4) (NOT (NTH 5 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 5))). Appealing to the lemma CAR-CDR-ELIM, we now replace V by (CONS X Z) to eliminate (CAR V) and (CDR V). This generates the goal: (IMPLIES (AND X (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 5 (CONS X Z))) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))). But this further simplifies, appealing to the lemma CDR-CONS, and expanding SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))). Eliminate the irrelevant term. This produces: (IMPLIES (AND (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), which we will finally name *1.1.1.1.1.1. Case 3. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NTH 5 (CDR V)) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))). This simplifies, applying the lemmas ADD1-EQUAL and SUB1-ADD1, and unfolding the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP V) (CAR V) (NTH 5 (CDR V)) (EQUAL (PICK-MOVE2 (CDR V)) 4) (NOT (NTH 5 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 5))). Applying the lemma CAR-CDR-ELIM, replace V by (CONS X Z) to eliminate (CAR V) and (CDR V). This produces: (IMPLIES (AND X (NTH 5 Z) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 5 (CONS X Z))) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), which further simplifies, rewriting with CDR-CONS, and unfolding the functions SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (NTH 5 Z) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), which has an irrelevant term in it. By eliminating the term we get: (IMPLIES (AND (NTH 5 Z) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), which we will name *1.1.1.1.1.2. Case 2. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (ENDP (CDR V)) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))). This simplifies, rewriting with the lemma ADD1-EQUAL, and expanding the definitions of NLISTP, PICK-MOVE2, NUMBERP, and ENDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NOT (LESSP (LENGTH (CDR V)) 6)) (EQUAL (PICK-MOVE2 V) 5) (NOT (NTH 5 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 6))). This simplifies, rewriting with ADD1-EQUAL and SUB1-ADD1, and unfolding the definitions of NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP V) (CAR V) (NOT (LESSP (LENGTH (CDR V)) 6)) (EQUAL (PICK-MOVE2 (CDR V)) 4) (NOT (NTH 5 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 5))). However this again simplifies, using linear arithmetic, to: T. So let us turn our attention to: (IMPLIES (AND (NTH 5 Z) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), named *1.1.1.1.1.2 above. Ah ha! This conjecture is subsumed by the subgoal we named *1.1.1.1.1.1 above. So we now return to: (IMPLIES (AND (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), which is formula *1.1.1.1.1.1 above. We will try to prove it by induction. There are three plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP Z) (p Z)) (IMPLIES (AND (NOT (NLISTP Z)) (EQUAL (CAR Z) F)) (p Z)) (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (p (CDR Z))) (p Z))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT Z) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to six new goals: Case 6. (IMPLIES (AND (NLISTP Z) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), which simplifies, opening up the definitions of NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP Z)) (EQUAL (CAR Z) F) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), which simplifies, expanding NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NOT (EQUAL (PICK-MOVE2 (CDR Z)) 4)) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))), which simplifies, rewriting with ADD1-EQUAL and SUB1-ADD1, and unfolding the definitions of NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP Z) (CAR Z) (NOT (EQUAL (PICK-MOVE2 (CDR Z)) 4)) (EQUAL (PICK-MOVE2 (CDR Z)) 3) (NOT (NTH 4 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 4))), which again simplifies, expanding EQUAL, to: (IMPLIES (AND (LISTP Z) (CAR Z) (EQUAL (PICK-MOVE2 (CDR Z)) 3) (NOT (NTH 4 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 4))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS X V) to eliminate (CAR Z) and (CDR Z). The result is the conjecture: (IMPLIES (AND X (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 4 (CONS X V))) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))). But this further simplifies, applying the lemma CDR-CONS, and opening up the functions SUB1, NUMBERP, EQUAL, and NTH, to the formula: (IMPLIES (AND X (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))). Eliminate the irrelevant term. This produces: (IMPLIES (AND (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), which we will finally name *1.1.1.1.1.1.1. Case 3. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NTH 4 (CDR Z)) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))). This simplifies, applying ADD1-EQUAL and SUB1-ADD1, and unfolding NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to the conjecture: (IMPLIES (AND (LISTP Z) (CAR Z) (NTH 4 (CDR Z)) (EQUAL (PICK-MOVE2 (CDR Z)) 3) (NOT (NTH 4 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 4))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS X V) to eliminate (CAR Z) and (CDR Z). The result is: (IMPLIES (AND X (NTH 4 V) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 4 (CONS X V))) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))). However this further simplifies, rewriting with CDR-CONS, and opening up the definitions of SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (NTH 4 V) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), which has an irrelevant term in it. By eliminating the term we get the new formula: (IMPLIES (AND (NTH 4 V) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), which we will name *1.1.1.1.1.1.2. Case 2. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (ENDP (CDR Z)) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))). This simplifies, rewriting with the lemma ADD1-EQUAL, and opening up the functions NLISTP, PICK-MOVE2, NUMBERP, and ENDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NOT (LESSP (LENGTH (CDR Z)) 5)) (EQUAL (PICK-MOVE2 Z) 4) (NOT (NTH 4 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 5))). This simplifies, applying ADD1-EQUAL and SUB1-ADD1, and unfolding NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to the conjecture: (IMPLIES (AND (LISTP Z) (CAR Z) (NOT (LESSP (LENGTH (CDR Z)) 5)) (EQUAL (PICK-MOVE2 (CDR Z)) 3) (NOT (NTH 4 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 4))). However this again simplifies, using linear arithmetic, to: T. So we now return to: (IMPLIES (AND (NTH 4 V) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), named *1.1.1.1.1.1.2 above. You probably did not notice, but this conjecture is subsumed by another subgoal awaiting our attention, namely *1.1.1.1.1.1.1 above. So let us turn our attention to: (IMPLIES (AND (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), named *1.1.1.1.1.1.1 above. We will try to prove it by induction. There are three plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP V) (p V)) (IMPLIES (AND (NOT (NLISTP V)) (EQUAL (CAR V) F)) (p V)) (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (p (CDR V))) (p V))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT V) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates six new conjectures: Case 6. (IMPLIES (AND (NLISTP V) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), which simplifies, unfolding the definitions of NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP V)) (EQUAL (CAR V) F) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), which simplifies, unfolding NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NOT (EQUAL (PICK-MOVE2 (CDR V)) 3)) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), which simplifies, applying ADD1-EQUAL and SUB1-ADD1, and unfolding the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP V) (CAR V) (NOT (EQUAL (PICK-MOVE2 (CDR V)) 3)) (EQUAL (PICK-MOVE2 (CDR V)) 2) (NOT (NTH 3 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 3))), which again simplifies, opening up the function EQUAL, to the conjecture: (IMPLIES (AND (LISTP V) (CAR V) (EQUAL (PICK-MOVE2 (CDR V)) 2) (NOT (NTH 3 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 3))). Appealing to the lemma CAR-CDR-ELIM, we now replace V by (CONS X Z) to eliminate (CAR V) and (CDR V). The result is the formula: (IMPLIES (AND X (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 3 (CONS X Z))) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))). But this further simplifies, appealing to the lemma CDR-CONS, and opening up the functions SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))). Eliminate the irrelevant term. We would thus like to prove: (IMPLIES (AND (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))), which we will finally name *1.1.1.1.1.1.1.1. Case 3. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NTH 3 (CDR V)) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))). This simplifies, rewriting with ADD1-EQUAL and SUB1-ADD1, and expanding the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP V) (CAR V) (NTH 3 (CDR V)) (EQUAL (PICK-MOVE2 (CDR V)) 2) (NOT (NTH 3 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 3))). Appealing to the lemma CAR-CDR-ELIM, we now replace V by (CONS X Z) to eliminate (CAR V) and (CDR V). We must thus prove: (IMPLIES (AND X (NTH 3 Z) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 3 (CONS X Z))) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))). This further simplifies, rewriting with CDR-CONS, and expanding the functions SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (NTH 3 Z) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))), which has an irrelevant term in it. By eliminating the term we get: (IMPLIES (AND (NTH 3 Z) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))), which we will name *1.1.1.1.1.1.1.2. Case 2. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (ENDP (CDR V)) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))). This simplifies, applying ADD1-EQUAL, and unfolding NLISTP, PICK-MOVE2, NUMBERP, and ENDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NOT (LESSP (LENGTH (CDR V)) 4)) (EQUAL (PICK-MOVE2 V) 3) (NOT (NTH 3 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 4))), which simplifies, applying ADD1-EQUAL and SUB1-ADD1, and unfolding the definitions of NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP V) (CAR V) (NOT (LESSP (LENGTH (CDR V)) 4)) (EQUAL (PICK-MOVE2 (CDR V)) 2) (NOT (NTH 3 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 3))), which again simplifies, using linear arithmetic, to: T. So we now return to: (IMPLIES (AND (NTH 3 Z) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))), which is formula *1.1.1.1.1.1.1.2 above. You probably did not notice, but this conjecture is subsumed by the subgoal we named *1.1.1.1.1.1.1.1 above. So we now return to: (IMPLIES (AND (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))), named *1.1.1.1.1.1.1.1 above. We will try to prove it by induction. The recursive terms in the conjecture suggest three inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP Z) (p Z)) (IMPLIES (AND (NOT (NLISTP Z)) (EQUAL (CAR Z) F)) (p Z)) (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (p (CDR Z))) (p Z))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to prove that the measure (COUNT Z) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to the following six new conjectures: Case 6. (IMPLIES (AND (NLISTP Z) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))). This simplifies, opening up NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP Z)) (EQUAL (CAR Z) F) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))). This simplifies, expanding NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NOT (EQUAL (PICK-MOVE2 (CDR Z)) 2)) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))). This simplifies, appealing to the lemmas ADD1-EQUAL and SUB1-ADD1, and opening up the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP Z) (CAR Z) (NOT (EQUAL (PICK-MOVE2 (CDR Z)) 2)) (EQUAL (PICK-MOVE2 (CDR Z)) 1) (NOT (NTH 2 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 2))), which again simplifies, expanding the function EQUAL, to: (IMPLIES (AND (LISTP Z) (CAR Z) (EQUAL (PICK-MOVE2 (CDR Z)) 1) (NOT (NTH 2 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 2))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS X V) to eliminate (CAR Z) and (CDR Z). We must thus prove the formula: (IMPLIES (AND X (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 2 (CONS X V))) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))). This further simplifies, rewriting with the lemma CDR-CONS, and opening up SUB1, NUMBERP, EQUAL, and NTH, to: (IMPLIES (AND X (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))). Eliminate the irrelevant term. We would thus like to prove: (IMPLIES (AND (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))), which we will finally name *1.1.1.1.1.1.1.1.1. Case 3. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NTH 2 (CDR Z)) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))). This simplifies, appealing to the lemmas ADD1-EQUAL and SUB1-ADD1, and unfolding the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP Z) (CAR Z) (NTH 2 (CDR Z)) (EQUAL (PICK-MOVE2 (CDR Z)) 1) (NOT (NTH 2 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 2))). Applying the lemma CAR-CDR-ELIM, replace Z by (CONS X V) to eliminate (CAR Z) and (CDR Z). This produces the new formula: (IMPLIES (AND X (NTH 2 V) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 2 (CONS X V))) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))), which further simplifies, applying CDR-CONS, and expanding the definitions of SUB1, NUMBERP, EQUAL, and NTH, to the new formula: (IMPLIES (AND X (NTH 2 V) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))), which has an irrelevant term in it. By eliminating the term we get: (IMPLIES (AND (NTH 2 V) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))), which we will name *1.1.1.1.1.1.1.1.2. Case 2. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (ENDP (CDR Z)) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))). This simplifies, applying ADD1-EQUAL, and opening up the definitions of NLISTP, PICK-MOVE2, NUMBERP, and ENDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (EQUAL (CAR Z) F)) (NOT (LESSP (LENGTH (CDR Z)) 3)) (EQUAL (PICK-MOVE2 Z) 2) (NOT (NTH 2 Z)) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 3))), which simplifies, applying the lemmas ADD1-EQUAL and SUB1-ADD1, and expanding the functions NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP Z) (CAR Z) (NOT (LESSP (LENGTH (CDR Z)) 3)) (EQUAL (PICK-MOVE2 (CDR Z)) 1) (NOT (NTH 2 Z)) (NOT (ENDP (CDR Z)))) (NOT (LESSP (LENGTH (CDR Z)) 2))). But this again simplifies, using linear arithmetic, to: T. So we now return to: (IMPLIES (AND (NTH 2 V) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))), named *1.1.1.1.1.1.1.1.2 above. You probably did not notice, but this conjecture is subsumed by another subgoal awaiting our attention, namely *1.1.1.1.1.1.1.1.1 above. So let us turn our attention to: (IMPLIES (AND (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))), which we named *1.1.1.1.1.1.1.1.1 above. Let us appeal to the induction principle. There are three plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP V) (p V)) (IMPLIES (AND (NOT (NLISTP V)) (EQUAL (CAR V) F)) (p V)) (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (p (CDR V))) (p V))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to establish that the measure (COUNT V) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to the following six new formulas: Case 6. (IMPLIES (AND (NLISTP V) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))). This simplifies, expanding the definitions of NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP V)) (EQUAL (CAR V) F) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))). This simplifies, opening up NLISTP, PICK-MOVE2, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NOT (EQUAL (PICK-MOVE2 (CDR V)) 1)) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))). This simplifies, applying the lemmas ADD1-EQUAL and SUB1-ADD1, and opening up the definitions of NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to the new formula: (IMPLIES (AND (LISTP V) (CAR V) (NOT (EQUAL (PICK-MOVE2 (CDR V)) 1)) (EQUAL (PICK-MOVE2 (CDR V)) 0) (NOT (NTH 1 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 1))), which again simplifies, expanding the function EQUAL, to the goal: (IMPLIES (AND (LISTP V) (CAR V) (EQUAL (PICK-MOVE2 (CDR V)) 0) (NOT (NTH 1 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 1))). Appealing to the lemma CAR-CDR-ELIM, we now replace V by (CONS X Z) to eliminate (CAR V) and (CDR V). This generates: (IMPLIES (AND X (EQUAL (PICK-MOVE2 Z) 0) (NOT (NTH 1 (CONS X Z))) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 1))). But this further simplifies, rewriting with the lemma CDR-CONS, and unfolding the definitions of SUB1, NUMBERP, EQUAL, NTH, and ENDP, to: (IMPLIES (AND X (EQUAL (PICK-MOVE2 Z) 0) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))). However this again simplifies, opening up the functions EQUAL and PICK-MOVE2, to: (IMPLIES (AND X (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))). Eliminate the irrelevant term. We thus obtain the new goal: (IMPLIES (AND (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))), which we will finally name *1.1.1.1.1.1.1.1.1.1. Case 3. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NTH 1 (CDR V)) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))). This simplifies, applying ADD1-EQUAL and SUB1-ADD1, and expanding the definitions of NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to the goal: (IMPLIES (AND (LISTP V) (CAR V) (NTH 1 (CDR V)) (EQUAL (PICK-MOVE2 (CDR V)) 0) (NOT (NTH 1 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 1))). Appealing to the lemma CAR-CDR-ELIM, we now replace V by (CONS X Z) to eliminate (CAR V) and (CDR V). We must thus prove: (IMPLIES (AND X (NTH 1 Z) (EQUAL (PICK-MOVE2 Z) 0) (NOT (NTH 1 (CONS X Z))) (NOT (ENDP Z))) (NOT (LESSP (LENGTH Z) 1))). This further simplifies, rewriting with the lemma CDR-CONS, and expanding the definitions of SUB1, NUMBERP, EQUAL, NTH, and ENDP, to the conjecture: (IMPLIES (AND X (NTH 1 Z) (EQUAL (PICK-MOVE2 Z) 0) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))). This again simplifies, expanding EQUAL and PICK-MOVE2, to: (IMPLIES (AND X (NTH 1 Z) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))). Eliminate the irrelevant term. We thus obtain the new formula: (IMPLIES (AND (NTH 1 Z) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))), which we will finally name *1.1.1.1.1.1.1.1.1.2. Case 2. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (ENDP (CDR V)) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))). This simplifies, applying ADD1-EQUAL, and opening up NLISTP, PICK-MOVE2, NUMBERP, and ENDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP V)) (NOT (EQUAL (CAR V) F)) (NOT (LESSP (LENGTH (CDR V)) 2)) (EQUAL (PICK-MOVE2 V) 1) (NOT (NTH 1 V)) (NOT (ENDP V))) (NOT (LESSP (LENGTH V) 2))), which simplifies, rewriting with the lemmas ADD1-EQUAL and SUB1-ADD1, and unfolding NLISTP, PICK-MOVE2, NUMBERP, ENDP, LENGTH, SUB1, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP V) (CAR V) (NOT (LESSP (LENGTH (CDR V)) 2)) (EQUAL (PICK-MOVE2 (CDR V)) 0) (NOT (NTH 1 V)) (NOT (ENDP (CDR V)))) (NOT (LESSP (LENGTH (CDR V)) 1))). This again simplifies, using linear arithmetic, to: T. So we now return to: (IMPLIES (AND (NTH 1 Z) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))), which is formula *1.1.1.1.1.1.1.1.1.2 above. What luck! This conjecture is subsumed by another subgoal awaiting our attention, namely *1.1.1.1.1.1.1.1.1.1 above. So next consider: (IMPLIES (AND (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))), which we named *1.1.1.1.1.1.1.1.1.1 above. We will try to prove it by induction. There is only one suggested induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP Z) (p Z)) (IMPLIES (AND (NOT (NLISTP Z)) (p (CDR Z))) (p Z))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT Z) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces four new formulas: Case 4. (IMPLIES (AND (NLISTP Z) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))), which simplifies, expanding the function NLISTP, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP Z)) (CADR Z) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))), which simplifies, rewriting with SUB1-ADD1, and unfolding the functions NLISTP, LENGTH, SUB1, NUMBERP, EQUAL, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (LISTP (CDR Z))) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))). This simplifies, applying SUB1-ADD1, and expanding the definitions of NLISTP, LENGTH, SUB1, NUMBERP, EQUAL, and LESSP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (LESSP (LENGTH (CDR Z)) 1)) (NOT (CAR Z)) (LISTP Z)) (NOT (LESSP (LENGTH Z) 1))), which simplifies, applying the lemma SUB1-ADD1, and expanding NLISTP, LENGTH, SUB1, NUMBERP, EQUAL, and LESSP, to: T. That finishes the proof of *1.1.1.1.1.1.1.1.1.1, which, in turn, finishes the proof of *1.1.1.1.1.1.1.1.1, which finishes the proof of *1.1.1.1.1.1.1.1, which finishes the proof of *1.1.1.1.1.1.1, which, consequently, finishes the proof of *1.1.1.1.1.1, which finishes the proof of *1.1.1.1.1, which finishes the proof of *1.1.1.1, which finishes the proof of *1.1.1, which, in turn, finishes the proof of *1.1, which finishes the proof of *1. Q.E.D. [ 0.0 0.2 0.1 ] NOT-ENDP-IMPLIES-LEGALP-PICK-MOVE2 (PROVE-LEMMA PICK-MOVE1-IS-PICK-MOVE2 (REWRITE) (IMPLIES (AND (NOT (ENDP BOARD)) (NUMBERP I)) (EQUAL (PICK-MOVE1 BOARD I) (PLUS I (PICK-MOVE2 BOARD))))) Give the conjecture the name *1. Perhaps we can prove it by induction. The recursive terms in the conjecture suggest four inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (p (CDR BOARD) (ADD1 I))) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD))) (p BOARD I))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for I. The above induction scheme leads to the following four new conjectures: Case 4. (IMPLIES (AND (NLISTP BOARD) (NOT (ENDP BOARD)) (NUMBERP I)) (EQUAL (PICK-MOVE1 BOARD I) (PLUS I (PICK-MOVE2 BOARD)))). This simplifies, opening up the functions NLISTP and ENDP, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (ENDP (CDR BOARD)) (NOT (ENDP BOARD)) (NUMBERP I)) (EQUAL (PICK-MOVE1 BOARD I) (PLUS I (PICK-MOVE2 BOARD)))). This simplifies, opening up the functions NLISTP and ENDP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (EQUAL (PICK-MOVE1 (CDR BOARD) (ADD1 I)) (PLUS (ADD1 I) (PICK-MOVE2 (CDR BOARD)))) (NOT (ENDP BOARD)) (NUMBERP I)) (EQUAL (PICK-MOVE1 BOARD I) (PLUS I (PICK-MOVE2 BOARD)))). This simplifies, applying SUB1-ADD1, and opening up the functions NLISTP, PLUS, ENDP, PICK-MOVE1, and PICK-MOVE2, to: (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (EQUAL (PICK-MOVE1 (CDR BOARD) (ADD1 I)) (ADD1 (PLUS I (PICK-MOVE2 (CDR BOARD))))) (NOT (ENDP (CDR BOARD))) (NUMBERP I)) (EQUAL (PICK-MOVE1 (CDR BOARD) (ADD1 I)) (PLUS I (ADD1 (PICK-MOVE2 (CDR BOARD)))))). However this again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (NOT (ENDP BOARD)) (NUMBERP I)) (EQUAL (PICK-MOVE1 BOARD I) (PLUS I (PICK-MOVE2 BOARD)))), which simplifies, expanding the definitions of NLISTP, ENDP, PICK-MOVE1, EQUAL, and PICK-MOVE2, to: (IMPLIES (AND (LISTP BOARD) (NOT (CAR BOARD)) (NUMBERP I)) (EQUAL I (PLUS I 0))). But this again simplifies, using linear arithmetic, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] PICK-MOVE1-IS-PICK-MOVE2 (COMPILE-UNCOMPILED-DEFNS "tmp") Loading ./basic/tmp.o Finished loading ./basic/tmp.o /v/hank/v28/boyer/nqthm-2nd/nqthm-1992/examples/basic/tmp.lisp (PROVE-LEMMA LEGALP-PICK-MOVE (REWRITE) (IMPLIES (AND (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10)) (LEGALP (PICK-MOVE BOARD) BOARD)) ((ENABLE PICK-MOVE) (DISABLE ASSOC))) WARNING: Note that the rewrite rule LEGALP-PICK-MOVE will be stored so as to apply only to terms with the nonrecursive function symbol LEGALP. This conjecture simplifies, rewriting with PICK-MOVE1-IS-PICK-MOVE2, and expanding the definitions of PLUS, EQUAL, and PICK-MOVE, to the following two new goals: Case 2. (IMPLIES (AND (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10) (NOT (ASSOC BOARD (LIST (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F 'X F 'O 'O F 'X 'X) 3) (CONS (LIST F F 'X F 'O 'O 'X F 'X) 3) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST F F F F F F F F 'X) 4) (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST F F F F F F F 'X F) 4) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST F 'O 'X F 'O F 'X F 'X) 7) (CONS (LIST F 'O 'X F 'O F 'X 'X F) 8) (CONS (LIST F 'O 'X F 'O 'X 'X F F) 7) (CONS (LIST F 'O 'X 'X 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST F F 'X F 'O F 'X F F) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST F F F F F F 'X F F) 4) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST F F F F F 'X F F F) 4) (CONS (LIST 'O 'X 'O F 'X F F F 'X) 7) (CONS (LIST 'O F F F 'X F F F 'X) 2) (CONS (LIST 'O 'O 'X F 'X F F 'X F) 6) (CONS (LIST 'O 'X 'O F 'X F 'X F F) 7) (CONS (LIST 'O F F F 'X F 'X F F) 2) (CONS (LIST 'O F F 'O 'X 'X F F 'X) 6) (CONS (LIST 'O F F 'O 'X 'X F 'X F) 6) (CONS (LIST 'O F F 'O 'X 'X 'X F F) 2) (CONS (LIST 'O F 'X 'O 'X 'X F F F) 6) (CONS (LIST 'O 'X F 'O 'X 'X F F F) 6) (CONS (LIST 'O F F F 'X 'X F F F) 3) (CONS (LIST 'O 'X 'O 'X 'X 'O 'X F F) 8) (CONS (LIST 'O F F 'X 'X 'O 'X F F) 2) (CONS (LIST 'O F 'X 'X 'X 'O F F F) 6) (CONS (LIST 'O 'X F 'X 'X 'O F F F) 7) (CONS (LIST 'O F F 'X 'X F F F F) 5) (CONS (LIST 'O F 'X F 'X F 'O F 'X) 3) (CONS (LIST 'O F 'X F 'X F 'O 'X F) 3) (CONS (LIST 'O F 'X F 'X 'X 'O F F) 3) (CONS (LIST 'O F 'X 'X 'X F 'O F F) 5) (CONS (LIST 'O F 'X F 'X F F F F) 6) (CONS (LIST 'O 'X F 'O 'X 'X F 'O 'X) 6) (CONS (LIST 'O 'X F F 'X 'X F 'O F) 3) (CONS (LIST 'O 'X F 'X 'X F F 'O F) 5) (CONS (LIST 'O 'X 'X F 'X 'X 'O 'O F) 8) (CONS (LIST 'O 'X 'X 'X 'X F 'O 'O F) 8) (CONS (LIST 'O 'X 'X F 'X F F 'O F) 6) (CONS (LIST 'O 'X F F 'X F F F F) 7) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST F F F 'X F F F F F) 4) (CONS (LIST F F 'X F 'O 'O F 'X 'X) 3) (CONS (LIST F F 'X F 'O 'O 'X F 'X) 3) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'O 'X F 'O F 'X F 'X) 7) (CONS (LIST F 'O 'X F 'O F 'X 'X F) 8) (CONS (LIST F 'O 'X F 'O 'X 'X F F) 7) (CONS (LIST F 'O 'X 'X 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST F F 'X F 'O F 'X F F) 1) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST F F 'X F F F F F F) 4) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST F 'X F F F F F F F) 4) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST 'X F F F F F F F F) 4))))) (LEGALP (PICK-MOVE2 BOARD) BOARD)). However this again simplifies, rewriting with the lemma NOT-ENDP-IMPLIES-LEGALP-PICK-MOVE2, to: T. Case 1. (IMPLIES (AND (NOT (ENDP BOARD)) (LESSP (LENGTH BOARD) 10) (ASSOC BOARD (LIST (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F 'X F 'O 'O F 'X 'X) 3) (CONS (LIST F F 'X F 'O 'O 'X F 'X) 3) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST F F F F F F F F 'X) 4) (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST F F F F F F F 'X F) 4) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST F 'O 'X F 'O F 'X F 'X) 7) (CONS (LIST F 'O 'X F 'O F 'X 'X F) 8) (CONS (LIST F 'O 'X F 'O 'X 'X F F) 7) (CONS (LIST F 'O 'X 'X 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST F F 'X F 'O F 'X F F) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST F F F F F F 'X F F) 4) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X 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'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST F F F F F 'X F F F) 4) (CONS (LIST 'O 'X 'O F 'X F F F 'X) 7) (CONS (LIST 'O F F F 'X F F F 'X) 2) (CONS (LIST 'O 'O 'X F 'X F F 'X F) 6) (CONS (LIST 'O 'X 'O F 'X F 'X F F) 7) (CONS (LIST 'O F F F 'X F 'X F F) 2) (CONS (LIST 'O F F 'O 'X 'X F F 'X) 6) (CONS (LIST 'O F F 'O 'X 'X F 'X F) 6) (CONS (LIST 'O F F 'O 'X 'X 'X F F) 2) (CONS (LIST 'O F 'X 'O 'X 'X F F F) 6) (CONS (LIST 'O 'X F 'O 'X 'X F F F) 6) (CONS (LIST 'O F F F 'X 'X F F F) 3) (CONS (LIST 'O 'X 'O 'X 'X 'O 'X F F) 8) (CONS (LIST 'O F F 'X 'X 'O 'X F F) 2) (CONS (LIST 'O F 'X 'X 'X 'O F F F) 6) (CONS (LIST 'O 'X F 'X 'X 'O F F F) 7) (CONS (LIST 'O F F 'X 'X F F F F) 5) (CONS (LIST 'O F 'X F 'X F 'O F 'X) 3) (CONS (LIST 'O F 'X F 'X F 'O 'X F) 3) (CONS (LIST 'O F 'X F 'X 'X 'O F F) 3) (CONS (LIST 'O F 'X 'X 'X F 'O F F) 5) (CONS (LIST 'O F 'X F 'X F F F F) 6) (CONS (LIST 'O 'X F 'O 'X 'X F 'O 'X) 6) (CONS (LIST 'O 'X F F 'X 'X F 'O F) 3) (CONS (LIST 'O 'X F 'X 'X F F 'O F) 5) (CONS (LIST 'O 'X 'X F 'X 'X 'O 'O F) 8) (CONS (LIST 'O 'X 'X 'X 'X F 'O 'O F) 8) (CONS (LIST 'O 'X 'X F 'X F F 'O F) 6) (CONS (LIST 'O 'X F F 'X F F F F) 7) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST F F F 'X F F F F F) 4) (CONS (LIST F F 'X F 'O 'O F 'X 'X) 3) (CONS (LIST F F 'X F 'O 'O 'X F 'X) 3) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'O 'X F 'O F 'X F 'X) 7) (CONS (LIST F 'O 'X F 'O F 'X 'X F) 8) (CONS (LIST F 'O 'X F 'O 'X 'X F F) 7) (CONS (LIST F 'O 'X 'X 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST F F 'X F 'O F 'X F F) 1) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST F F 'X F F F F F F) 4) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST F 'X F F F F F F F) 4) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST 'X F F F F F F F F) 4)))) BOARD)), which again simplifies, rewriting with LEGAL-BOOKP-IMPLIES-LEGALP-CDR-ASSOC, and unfolding the definition of LEGAL-BOOKP, to: T. Q.E.D. [ 0.0 0.0 0.1 ] LEGALP-PICK-MOVE (PROVE-LEMMA BOARDP-MOVE (REWRITE) (IMPLIES (AND (BOARDP BOARD) (OR (EQUAL PLAYER 'X) (EQUAL PLAYER 'O))) (BOARDP (MOVE PLAYER I BOARD)))) This conjecture simplifies, expanding the function OR, to the following two new formulas: Case 2. (IMPLIES (AND (BOARDP BOARD) (EQUAL PLAYER 'X)) (BOARDP (MOVE 'X I BOARD))). This again simplifies, clearly, to the new conjecture: (IMPLIES (BOARDP BOARD) (BOARDP (MOVE 'X I BOARD))), which we will name *1. Case 1. (IMPLIES (AND (BOARDP BOARD) (EQUAL PLAYER 'O)) (BOARDP (MOVE 'O I BOARD))). This again simplifies, obviously, to: (IMPLIES (BOARDP BOARD) (BOARDP (MOVE 'O I BOARD))), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (AND (IMPLIES (AND (BOARDP BOARD) (EQUAL PLAYER 'O)) (BOARDP (MOVE PLAYER I BOARD))) (IMPLIES (AND (BOARDP BOARD) (EQUAL PLAYER 'X)) (BOARDP (MOVE PLAYER I BOARD)))). We named this *1. We will try to prove it by induction. Six inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p PLAYER I BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I)) (p PLAYER I BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (p PLAYER (SUB1 I) (CDR BOARD))) (p PLAYER I BOARD))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP inform us that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme leads to eight new goals: Case 8. (IMPLIES (AND (NLISTP BOARD) (BOARDP BOARD) (EQUAL PLAYER 'O)) (BOARDP (MOVE PLAYER I BOARD))), which simplifies, expanding NLISTP, BOARDP, LISTP, and MOVE, to: T. Case 7. (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I) (BOARDP BOARD) (EQUAL PLAYER 'O)) (BOARDP (MOVE PLAYER I BOARD))), which simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding NLISTP, ZEROP, BOARDP, EQUAL, and MOVE, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (BOARDP (CDR BOARD))) (BOARDP BOARD) (EQUAL PLAYER 'O)) (BOARDP (MOVE PLAYER I BOARD))). This simplifies, unfolding NLISTP, ZEROP, and BOARDP, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (BOARDP (MOVE PLAYER (SUB1 I) (CDR BOARD))) (BOARDP BOARD) (EQUAL PLAYER 'O)) (BOARDP (MOVE PLAYER I BOARD))). This simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up the functions NLISTP, ZEROP, BOARDP, MOVE, and EQUAL, to: T. Case 4. (IMPLIES (AND (NLISTP BOARD) (BOARDP BOARD) (EQUAL PLAYER 'X)) (BOARDP (MOVE PLAYER I BOARD))). This simplifies, unfolding the functions NLISTP, BOARDP, LISTP, and MOVE, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I) (BOARDP BOARD) (EQUAL PLAYER 'X)) (BOARDP (MOVE PLAYER I BOARD))). This simplifies, applying CDR-CONS and CAR-CONS, and expanding NLISTP, ZEROP, BOARDP, EQUAL, and MOVE, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (BOARDP (CDR BOARD))) (BOARDP BOARD) (EQUAL PLAYER 'X)) (BOARDP (MOVE PLAYER I BOARD))), which simplifies, unfolding the functions NLISTP, ZEROP, and BOARDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (BOARDP (MOVE PLAYER (SUB1 I) (CDR BOARD))) (BOARDP BOARD) (EQUAL PLAYER 'X)) (BOARDP (MOVE PLAYER I BOARD))), which simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding NLISTP, ZEROP, BOARDP, MOVE, and EQUAL, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] BOARDP-MOVE (PROVE-LEMMA LEGALP-IMPLIES-LESSP-9 (REWRITE) (IMPLIES (AND (LEGALP (CDR (ASSOC BOARD ALIST)) BOARD) (EQUAL (LENGTH BOARD) 9)) (LESSP (CDR (ASSOC BOARD ALIST)) 9)) ((ENABLE LEGALP))) WARNING: Note that the proposed lemma LEGALP-IMPLIES-LESSP-9 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations LEGALP, AND, and IMPLIES, to the new formula: T, which simplifies, obviously, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP-IMPLIES-LESSP-9 (PROVE-LEMMA ENDP-IMPLIES-PICK-MOVE1-0 (REWRITE) (IMPLIES (ENDP BOARD) (EQUAL (PICK-MOVE1 BOARD I) 0))) Name the conjecture *1. We will appeal to induction. The recursive terms in the conjecture suggest two inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (p (CDR BOARD) (ADD1 I))) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD))) (p BOARD I))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for I. The above induction scheme produces the following four new formulas: Case 4. (IMPLIES (AND (NLISTP BOARD) (ENDP BOARD)) (EQUAL (PICK-MOVE1 BOARD I) 0)). This simplifies, expanding the definitions of NLISTP, ENDP, PICK-MOVE1, and EQUAL, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (NOT (ENDP (CDR BOARD))) (ENDP BOARD)) (EQUAL (PICK-MOVE1 BOARD I) 0)). This simplifies, expanding the definitions of NLISTP and ENDP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (EQUAL (PICK-MOVE1 (CDR BOARD) (ADD1 I)) 0) (ENDP BOARD)) (EQUAL (PICK-MOVE1 BOARD I) 0)). This simplifies, opening up the functions NLISTP, ENDP, PICK-MOVE1, and EQUAL, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (ENDP BOARD)) (EQUAL (PICK-MOVE1 BOARD I) 0)). This simplifies, opening up the functions NLISTP and ENDP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] ENDP-IMPLIES-PICK-MOVE1-0 (PROVE-LEMMA LESSP-PICK-MOVE (REWRITE) (IMPLIES (EQUAL (LENGTH BOARD) 9) (LESSP (PICK-MOVE BOARD) 9)) ((DISABLE LEGALP-PICK-MOVE ASSOC) (ENABLE PICK-MOVE) (USE (LEGALP-PICK-MOVE)))) WARNING: Note that the proposed lemma LESSP-PICK-MOVE is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula simplifies, applying ENDP-IMPLIES-PICK-MOVE1-0, and expanding the functions NOT, LESSP, AND, PICK-MOVE, and IMPLIES, to four new goals: Case 4. (IMPLIES (AND (ENDP BOARD) (EQUAL (LENGTH BOARD) 9) (NOT (ASSOC BOARD (LIST (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O 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'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O 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(CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 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(CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X 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(CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST F F 'X F F F F F F) 4) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST F 'X F F F F F F F) 4) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST 'X F F F F F F F F) 4))))) (LESSP 0 9)), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (ENDP BOARD) (EQUAL (LENGTH BOARD) 9) (ASSOC BOARD (LIST (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F 'X F 'O 'O F 'X 'X) 3) (CONS (LIST F F 'X F 'O 'O 'X F 'X) 3) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST F F F F F F F F 'X) 4) (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST F F F F F F F 'X F) 4) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST F 'O 'X F 'O F 'X F 'X) 7) (CONS (LIST F 'O 'X F 'O F 'X 'X F) 8) (CONS (LIST F 'O 'X F 'O 'X 'X F F) 7) (CONS (LIST F 'O 'X 'X 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST F F 'X F 'O F 'X F F) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST F F F F F F 'X F F) 4) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST F F F F F 'X F F F) 4) (CONS (LIST 'O 'X 'O F 'X F F F 'X) 7) (CONS (LIST 'O F F F 'X F F F 'X) 2) (CONS (LIST 'O 'O 'X F 'X F F 'X F) 6) (CONS (LIST 'O 'X 'O F 'X F 'X F F) 7) (CONS (LIST 'O F F F 'X F 'X F F) 2) (CONS (LIST 'O F F 'O 'X 'X F F 'X) 6) (CONS (LIST 'O F F 'O 'X 'X F 'X F) 6) (CONS (LIST 'O F F 'O 'X 'X 'X F F) 2) (CONS (LIST 'O F 'X 'O 'X 'X F F F) 6) (CONS (LIST 'O 'X F 'O 'X 'X F F F) 6) (CONS (LIST 'O F F F 'X 'X F F F) 3) (CONS (LIST 'O 'X 'O 'X 'X 'O 'X F F) 8) (CONS (LIST 'O F F 'X 'X 'O 'X F F) 2) (CONS (LIST 'O F 'X 'X 'X 'O F F F) 6) (CONS (LIST 'O 'X F 'X 'X 'O F F F) 7) (CONS (LIST 'O F F 'X 'X F F F F) 5) (CONS (LIST 'O F 'X F 'X F 'O F 'X) 3) (CONS (LIST 'O F 'X F 'X F 'O 'X F) 3) (CONS (LIST 'O F 'X F 'X 'X 'O F F) 3) (CONS (LIST 'O F 'X 'X 'X F 'O F F) 5) (CONS (LIST 'O F 'X F 'X F F F F) 6) (CONS (LIST 'O 'X F 'O 'X 'X F 'O 'X) 6) (CONS (LIST 'O 'X F F 'X 'X F 'O F) 3) (CONS (LIST 'O 'X F 'X 'X F F 'O F) 5) (CONS (LIST 'O 'X 'X F 'X 'X 'O 'O F) 8) (CONS (LIST 'O 'X 'X 'X 'X F 'O 'O F) 8) (CONS (LIST 'O 'X 'X F 'X F F 'O F) 6) (CONS (LIST 'O 'X F F 'X F F F F) 7) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X 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F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'O 'X F 'O F 'X F 'X) 7) (CONS (LIST F 'O 'X F 'O F 'X 'X F) 8) (CONS (LIST F 'O 'X F 'O 'X 'X F F) 7) (CONS (LIST F 'O 'X 'X 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST F F 'X F 'O F 'X F F) 1) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST F F 'X F F F F F F) 4) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST F 'X F F F F F F F) 4) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST 'X F F F F F F F F) 4)))) 9)), which again simplifies, using linear arithmetic, applying LEGALP-IMPLIES-LESSP-9 and LEGAL-BOOKP-IMPLIES-LEGALP-CDR-ASSOC, and opening up the function LEGAL-BOOKP, to: T. Case 2. (IMPLIES (AND (NOT (ASSOC BOARD (LIST (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F 'X F 'O 'O F 'X 'X) 3) (CONS (LIST F F 'X F 'O 'O 'X F 'X) 3) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST F F F F F F F F 'X) 4) (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST F F F F F F F 'X F) 4) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST F 'O 'X F 'O F 'X F 'X) 7) (CONS (LIST F 'O 'X F 'O F 'X 'X F) 8) (CONS (LIST F 'O 'X F 'O 'X 'X F F) 7) (CONS (LIST F 'O 'X 'X 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST F F 'X F 'O F 'X F F) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST F F F F F F 'X F F) 4) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST F F F F F 'X F F F) 4) (CONS (LIST 'O 'X 'O F 'X F F F 'X) 7) (CONS (LIST 'O F F F 'X F F F 'X) 2) (CONS (LIST 'O 'O 'X F 'X F F 'X F) 6) (CONS (LIST 'O 'X 'O F 'X F 'X F F) 7) (CONS (LIST 'O F F F 'X F 'X F F) 2) (CONS (LIST 'O F F 'O 'X 'X F F 'X) 6) (CONS (LIST 'O F F 'O 'X 'X F 'X F) 6) (CONS (LIST 'O F F 'O 'X 'X 'X F F) 2) (CONS (LIST 'O F 'X 'O 'X 'X F F F) 6) (CONS (LIST 'O 'X F 'O 'X 'X F F F) 6) (CONS (LIST 'O F F F 'X 'X F F F) 3) (CONS (LIST 'O 'X 'O 'X 'X 'O 'X F F) 8) (CONS (LIST 'O F F 'X 'X 'O 'X F F) 2) (CONS (LIST 'O F 'X 'X 'X 'O F F F) 6) (CONS (LIST 'O 'X F 'X 'X 'O F F F) 7) (CONS (LIST 'O F F 'X 'X F F F F) 5) (CONS (LIST 'O F 'X F 'X F 'O F 'X) 3) (CONS (LIST 'O F 'X F 'X F 'O 'X F) 3) (CONS (LIST 'O F 'X F 'X 'X 'O F F) 3) (CONS (LIST 'O F 'X 'X 'X F 'O F F) 5) (CONS (LIST 'O F 'X F 'X F F F F) 6) (CONS (LIST 'O 'X F 'O 'X 'X F 'O 'X) 6) (CONS (LIST 'O 'X F F 'X 'X F 'O F) 3) (CONS (LIST 'O 'X F 'X 'X F F 'O F) 5) (CONS (LIST 'O 'X 'X F 'X 'X 'O 'O F) 8) (CONS (LIST 'O 'X 'X 'X 'X F 'O 'O F) 8) (CONS (LIST 'O 'X 'X F 'X F F 'O F) 6) (CONS (LIST 'O 'X F F 'X F F F F) 7) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O F F 'X 'O 'X F F 'X) 2) (CONS (LIST 'O F F 'X 'O 'X F 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST F F F 'X F F F F F) 4) (CONS (LIST F F 'X F 'O 'O F 'X 'X) 3) (CONS (LIST F F 'X F 'O 'O 'X F 'X) 3) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'O 'X F 'O F 'X F 'X) 7) (CONS (LIST F 'O 'X F 'O F 'X 'X F) 8) (CONS (LIST F 'O 'X F 'O 'X 'X F F) 7) (CONS (LIST F 'O 'X 'X 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST F F 'X F 'O F 'X F F) 1) (CONS (LIST 'X 'O 'X F 'O 'X 'X F 'O) 7) (CONS (LIST 'X 'O 'X 'X 'O 'X F F 'O) 7) (CONS (LIST F F 'X F 'O 'X F F F) 8) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST 'O F 'X 'X 'O F F F 'X) 5) (CONS (LIST 'O F 'X 'X 'O F F 'X F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST F F 'X F F F F F F) 4) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST 'O 'X F F 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F F 'O F 'X 'X F) 8) (CONS (LIST 'O 'X F F 'O 'X 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST 'O 'X 'O F 'O 'X 'X F 'X) 7) (CONS (LIST 'O 'X 'O F 'O 'X 'X 'X F) 8) (CONS (LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST F 'X F F F F F F F) 4) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST 'X F F F F F F F F) 4)))) (LEGALP (PICK-MOVE1 BOARD 0) BOARD) (EQUAL (LENGTH BOARD) 9)) (LESSP (PICK-MOVE1 BOARD 0) 9)). But this again simplifies, opening up the definition of LEGALP, to: T. Case 1. (IMPLIES (AND (ASSOC BOARD (LIST (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST F F 'O F 'O 'X F 'X 'X) 6) (CONS (LIST F F 'O 'X 'O 'X 'X 'O 'X) 1) (CONS (LIST F F 'O F 'O 'X 'X F 'X) 7) (CONS (LIST F F 'O 'X 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST F F F F 'O 'X F F 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O F 'X) 2) (CONS (LIST F F 'X 'X 'O F 'O F 'X) 5) (CONS (LIST F 'X F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST F F F 'X 'O F F F 'X) 6) (CONS (LIST F F 'X F 'O 'O F 'X 'X) 3) (CONS (LIST F F 'X F 'O 'O 'X F 'X) 3) (CONS (LIST 'O F 'X 'X 'O 'O F 'X 'X) 6) (CONS (LIST 'O F 'X 'X 'O 'O 'X F 'X) 7) (CONS (LIST F 'X 'X F 'O 'O F F 'X) 3) (CONS (LIST 'X F 'X F 'O 'O F F 'X) 3) (CONS (LIST F F 'X F 'O F F F 'X) 5) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O F 'O F 'X F 'X) 7) (CONS (LIST F 'X 'O F 'O 'X F F 'X) 6) (CONS (LIST F 'X 'O 'X 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST F 'X F F 'O F F F 'X) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST F F F F F F F F 'X) 4) (CONS (LIST F F F F 'O 'X 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F 'X 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST 'X F 'X F 'O 'O 'O 'X 'X) 3) (CONS (LIST F F 'X F 'O F 'O 'X 'X) 5) (CONS (LIST F 'X F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST F F F F 'O F F 'X 'X) 6) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X 'O F F 'O 'X 'X 'X 'O) 3) (CONS (LIST 'X 'O F 'X 'O 'X F 'X 'O) 6) (CONS (LIST F F F F 'O 'X F 'X F) 8) (CONS (LIST F F F 'X 'O F 'O 'X 'X) 2) (CONS (LIST F F F 'X 'O 'X 'O 'X F) 2) (CONS (LIST 'O F 'X 'X 'O F 'O 'X 'X) 5) (CONS (LIST 'O F 'X 'X 'O 'X 'O 'X F) 8) (CONS (LIST 'O 'X 'X 'X 'O F 'O 'X F) 8) (CONS (LIST F 'X F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST F F F 'X 'O F F 'X F) 6) (CONS (LIST 'X 'O 'X 'X 'O F F 'X 'O) 6) (CONS (LIST F F 'X F 'O F F 'X F) 8) (CONS (LIST F 'X 'O F 'O F F 'X 'X) 6) (CONS (LIST F 'X 'O 'X 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X 'O F 'O F 'X 'X 'O) 5) (CONS (LIST F 'X 'O F 'O F 'X 'X F) 8) (CONS (LIST F 'X 'O F 'O 'X F 'X F) 6) (CONS (LIST F 'X 'O 'X 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST F 'X F F 'O F F 'X F) 2) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST F F F F F F F 'X F) 4) (CONS (LIST F F F F 'O 'X 'X 'O 'X) 1) (CONS (LIST F F F 'X 'O F 'X 'O 'X) 1) (CONS (LIST F F 'X F 'O F 'X 'O 'X) 1) (CONS (LIST 'O 'X 'X F 'O F 'X 'O 'X) 5) (CONS (LIST F F F F 'O F 'X F 'X) 7) (CONS (LIST 'X F 'X 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X 'X F 'O 'O F 'X 'X 'O) 5) (CONS (LIST 'X F F F 'O F 'X 'X 'O) 3) (CONS (LIST F F F F 'O F 'X 'X F) 8) (CONS (LIST 'X F F F 'O 'X 'X F 'O) 3) (CONS (LIST F F F F 'O 'X 'X F F) 8) (CONS (LIST 'O F F 'X 'O F 'X F 'X) 7) (CONS (LIST 'O F F 'X 'O F 'X 'X F) 8) (CONS (LIST 'O F F 'X 'O 'X 'X F F) 8) (CONS (LIST 'O F 'X 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 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(LIST 'O 'X 'O 'X 'O 'X 'X F F) 8) (CONS (LIST F 'X 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST F 'X F F 'O 'X F F F) 2) (CONS (LIST 'O 'X 'O 'X 'O F F 'X 'X) 6) (CONS (LIST 'O 'X 'O 'X 'O F 'X F 'X) 7) (CONS (LIST 'O 'X F 'X 'O F F 'X F) 8) (CONS (LIST 'O 'X F 'X 'O F 'X F F) 8) (CONS (LIST 'O 'X F 'X 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'O 'X 'X F 'O F F F 'X) 5) (CONS (LIST 'O 'X 'X F 'O F F 'X F) 8) (CONS (LIST 'O 'X 'X F 'O F 'X F F) 8) (CONS (LIST 'O 'X 'X F 'O 'X F F F) 8) (CONS (LIST 'O 'X 'X 'X 'O F F F F) 8) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST F 'X F F F F F F F) 4) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O F F 'O F F 'X 'X) 6) (CONS (LIST 'X 'O F F 'O F 'X F 'X) 7) (CONS (LIST 'X 'O F F 'O 'X F F 'X) 7) (CONS (LIST 'X 'O F 'X 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X F F F 'O F 'O 'X 'X) 2) (CONS (LIST 'X F F F 'O 'X 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X 'O 'X F 'O F 'O 'X 'X) 5) (CONS (LIST 'X 'O 'X F 'O 'X 'O 'X F) 8) (CONS (LIST 'X F F F 'O F F 'X F) 6) (CONS (LIST 'X F F 'O 'O F 'X F 'X) 5) (CONS (LIST 'X F F 'O 'O F 'X 'X F) 5) (CONS (LIST 'X 'O F 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X 'O F 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'X 'O 'O F 'X F F) 5) (CONS (LIST 'X 'X F 'O 'O F 'X F F) 5) (CONS (LIST 'X F F F 'O F 'X F F) 3) (CONS (LIST 'X F 'O F 'O 'X F F 'X) 6) (CONS (LIST 'X F 'O F 'O 'X F 'X F) 6) (CONS (LIST 'X F 'O 'O 'O 'X 'X F 'X) 7) (CONS (LIST 'X F 'O 'O 'O 'X 'X 'X F) 8) (CONS (LIST 'X F 'O F 'O 'X 'X F F) 3) (CONS (LIST 'X F 'O 'X 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X F F F 'O 'X F F F) 2) (CONS (LIST 'X F F 'X 'O F 'O F 'X) 2) (CONS (LIST 'X F F 'X 'O F 'O 'X F) 2) (CONS (LIST 'X F F 'X 'O 'X 'O F F) 2) (CONS (LIST 'X 'O 'X 'X 'O F 'O F 'X) 7) (CONS (LIST 'X F F 'X 'O F F F F) 6) (CONS (LIST 'X 'O 'X F 'O F F F 'X) 7) (CONS (LIST 'X 'O 'X 'O 'O 'X F 'X F) 8) (CONS (LIST 'X 'O 'X F 'O F 'X F F) 7) (CONS (LIST 'X 'O 'X F 'O 'X F F F) 7) (CONS (LIST 'X 'O 'X 'X 'O F F F F) 7) (CONS (LIST 'X 'X 'O F 'O F F F 'X) 6) (CONS (LIST 'X 'X 'O F 'O F F 'X F) 6) (CONS (LIST 'X 'X 'O F 'O 'X F F F) 6) (CONS (LIST 'X 'X 'O 'X 'O F F F F) 6) (CONS (LIST 'X F F F F F F F F) 4)))) 9)), which again simplifies, using linear arithmetic and rewriting with LEGALP-IMPLIES-LESSP-9, to: T. Q.E.D. [ 0.0 0.2 0.2 ] LESSP-PICK-MOVE (PROVE-LEMMA TIC-TAC-TOE1-OPENER (REWRITE) (EQUAL (TIC-TAC-TOE1 XMOVES (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)) (IF (LEGALP (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)) (COND ((ENDP (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8))) (LIST (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)) 'DRAW)) ((WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8))) (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)))) (LIST (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8))) (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8))) 'O-WIN)) (T (CONS (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8))) (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8))) (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)))))))) (LIST (MOVE 'X (CAR XMOVES) (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)) 'O-WIN))) ((DISABLE LEGALP MOVE))) This simplifies, unfolding the function TIC-TAC-TOE1, to: T. Q.E.D. [ 0.0 0.0 0.0 ] TIC-TAC-TOE1-OPENER (DISABLE TIC-TAC-TOE1) [ 0.0 0.0 0.0 ] TIC-TAC-TOE1-OFF (DEFN LEGALP1 (X BOARD) (IF (ZEROP X) (NOT (CAR BOARD)) (IF (NLISTP BOARD) F (LEGALP1 (SUB1 X) (CDR BOARD))))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of NLISTP and ZEROP can be used to show that the measure (COUNT X) decreases according to the well-founded relation LESSP in each recursive call. Hence, LEGALP1 is accepted under the definitional principle. The definition of LEGALP1 can be justified in another way. Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definitions of NLISTP and ZEROP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Note that: (OR (FALSEP (LEGALP1 X BOARD)) (TRUEP (LEGALP1 X BOARD))) is a theorem. [ 0.0 0.0 0.0 ] LEGALP1 (PROVE-LEMMA LEGALP1-FACT1 (REWRITE) (IMPLIES (NOT (LESSP X (LENGTH BOARD))) (NOT (LEGALP1 X BOARD)))) Call the conjecture *1. Perhaps we can prove it by induction. Four inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (p X BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL X 0) (NOT (NUMBERP X)))) (p X BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (p (SUB1 X) (CDR BOARD))) (p X BOARD))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT can be used to prove that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme leads to four new goals: Case 4. (IMPLIES (AND (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (NOT (LESSP X (LENGTH BOARD)))) (NOT (LEGALP1 X BOARD))), which simplifies, applying the lemma CAR-NLISTP, and opening up the definitions of LENGTH, NOT, OR, EQUAL, LESSP, and LEGALP1, to three new formulas: Case 4.3. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (NOT (EQUAL X 0)) (NUMBERP X) (LISTP BOARD)) (NOT (LEGALP1 (SUB1 X) (CDR BOARD)))), which again simplifies, using linear arithmetic, to: T. Case 4.2. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (EQUAL X 0)) (CAR BOARD)), which again simplifies, using linear arithmetic, to: T. Case 4.1. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (NOT (NUMBERP X))) (CAR BOARD)), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL X 0) (NOT (NUMBERP X))) (NOT (LESSP X (LENGTH BOARD)))) (NOT (LEGALP1 X BOARD))), which simplifies, opening up the definitions of LENGTH, NOT, OR, EQUAL, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (LESSP (SUB1 X) (LENGTH (CDR BOARD))) (NOT (LESSP X (LENGTH BOARD)))) (NOT (LEGALP1 X BOARD))), which simplifies, applying SUB1-ADD1, and opening up the functions LENGTH, NOT, OR, and LESSP, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (LEGALP1 (SUB1 X) (CDR BOARD))) (NOT (LESSP X (LENGTH BOARD)))) (NOT (LEGALP1 X BOARD))). This simplifies, rewriting with SUB1-ADD1, and expanding the definitions of LENGTH, NOT, OR, LESSP, and LEGALP1, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP1-FACT1 (PROVE-LEMMA LEGALP1-FACT2 (REWRITE) (IMPLIES (NTH X BOARD) (NOT (LEGALP1 X BOARD)))) Name the conjecture *1. Perhaps we can prove it by induction. There are three plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP X) (p X BOARD)) (IMPLIES (AND (NOT (ZEROP X)) (NLISTP BOARD)) (p X BOARD)) (IMPLIES (AND (NOT (ZEROP X)) (NOT (NLISTP BOARD)) (p (SUB1 X) (CDR BOARD))) (p X BOARD))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of NLISTP and ZEROP establish that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme produces the following four new goals: Case 4. (IMPLIES (AND (ZEROP X) (NTH X BOARD)) (NOT (LEGALP1 X BOARD))). This simplifies, expanding ZEROP, EQUAL, NTH, and LEGALP1, to: T. Case 3. (IMPLIES (AND (NOT (ZEROP X)) (NLISTP BOARD) (NTH X BOARD)) (NOT (LEGALP1 X BOARD))). This simplifies, using linear arithmetic, rewriting with LEGALP1-FACT1, and unfolding the definitions of ZEROP, NLISTP, NTH, and LENGTH, to: T. Case 2. (IMPLIES (AND (NOT (ZEROP X)) (NOT (NLISTP BOARD)) (NOT (NTH (SUB1 X) (CDR BOARD))) (NTH X BOARD)) (NOT (LEGALP1 X BOARD))), which simplifies, unfolding the definitions of ZEROP, NLISTP, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP X)) (NOT (NLISTP BOARD)) (NOT (LEGALP1 (SUB1 X) (CDR BOARD))) (NTH X BOARD)) (NOT (LEGALP1 X BOARD))), which simplifies, opening up ZEROP, NLISTP, NTH, and LEGALP1, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP1-FACT2 (PROVE-LEMMA LEGALP1-FACT3 (REWRITE) (IMPLIES (AND (LESSP X (LENGTH BOARD)) (NOT (NTH X BOARD))) (LEGALP1 X BOARD))) Name the conjecture *1. Perhaps we can prove it by induction. Five inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (p X BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL X 0) (NOT (NUMBERP X)))) (p X BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (p (SUB1 X) (CDR BOARD))) (p X BOARD))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT inform us that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme leads to the following five new conjectures: Case 5. (IMPLIES (AND (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (LESSP X (LENGTH BOARD)) (NOT (NTH X BOARD))) (LEGALP1 X BOARD)). This simplifies, opening up LENGTH, NOT, OR, EQUAL, and LESSP, to: T. Case 4. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL X 0) (NOT (NUMBERP X))) (LESSP X (LENGTH BOARD)) (NOT (NTH X BOARD))) (LEGALP1 X BOARD)). This simplifies, unfolding the definitions of LENGTH, NOT, OR, EQUAL, LESSP, NTH, and LEGALP1, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (LESSP (SUB1 X) (LENGTH (CDR BOARD)))) (LESSP X (LENGTH BOARD)) (NOT (NTH X BOARD))) (LEGALP1 X BOARD)). This simplifies, applying SUB1-ADD1, and unfolding LENGTH, NOT, OR, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NTH (SUB1 X) (CDR BOARD)) (LESSP X (LENGTH BOARD)) (NOT (NTH X BOARD))) (LEGALP1 X BOARD)), which simplifies, applying the lemma SUB1-ADD1, and expanding LENGTH, NOT, OR, LESSP, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (LEGALP1 (SUB1 X) (CDR BOARD)) (LESSP X (LENGTH BOARD)) (NOT (NTH X BOARD))) (LEGALP1 X BOARD)), which simplifies, applying SUB1-ADD1, and opening up the functions LENGTH, NOT, OR, LESSP, NTH, and LEGALP1, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP1-FACT3 (PROVE-LEMMA LEGALP-IS-LEGALP1 (REWRITE) (IMPLIES (EQUAL (LENGTH BOARD) '9) (EQUAL (LEGALP X BOARD) (LEGALP1 X BOARD)))) WARNING: Note that the rewrite rule LEGALP-IS-LEGALP1 will be stored so as to apply only to terms with the nonrecursive function symbol LEGALP. This conjecture simplifies, expanding the definition of LEGALP, to the following three new formulas: Case 3. (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (NTH X BOARD)) (EQUAL F (LEGALP1 X BOARD))). This again simplifies, applying LEGALP1-FACT2, and opening up EQUAL, to: T. Case 2. (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (NOT (LESSP X 9))) (EQUAL F (LEGALP1 X BOARD))). This again simplifies, using linear arithmetic, applying LEGALP1-FACT1, and opening up the function EQUAL, to: T. Case 1. (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (LESSP X 9) (NOT (NTH X BOARD))) (EQUAL T (LEGALP1 X BOARD))). However this again simplifies, using linear arithmetic, rewriting with LEGALP1-FACT3, and expanding the function EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP-IS-LEGALP1 (DEFN LEGALP2 (X BOARD I) (IF (NLISTP BOARD) F (IF (AND (NOT (CAR BOARD)) (EQUAL (FIX X) I)) T (LEGALP2 X (CDR BOARD) (ADD1 I))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definitions of AND, FIX, NOT, and NLISTP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Hence, LEGALP2 is accepted under the principle of definition. Observe that (OR (FALSEP (LEGALP2 X BOARD I)) (TRUEP (LEGALP2 X BOARD I))) is a theorem. [ 0.0 0.0 0.0 ] LEGALP2 (PROVE-LEMMA LEGALP2-FACT1 (REWRITE) (IMPLIES (LESSP X I) (NOT (LEGALP2 X BOARD I)))) Give the conjecture the name *1. Let us appeal to the induction principle. Three inductions are suggested by terms in the conjecture. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p X BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (AND (NOT (CAR BOARD)) (EQUAL (FIX X) I))) (p X BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (AND (NOT (CAR BOARD)) (EQUAL (FIX X) I))) (p X (CDR BOARD) (ADD1 I))) (p X BOARD I))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definitions of AND, FIX, NOT, and NLISTP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for I. The above induction scheme generates four new conjectures: Case 4. (IMPLIES (AND (NLISTP BOARD) (LESSP X I)) (NOT (LEGALP2 X BOARD I))), which simplifies, expanding the functions NLISTP and LEGALP2, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (AND (NOT (CAR BOARD)) (EQUAL (FIX X) I)) (LESSP X I)) (NOT (LEGALP2 X BOARD I))), which simplifies, opening up the functions NLISTP, NOT, FIX, AND, LEGALP2, EQUAL, and LESSP, to: (IMPLIES (AND (LISTP BOARD) (NOT (CAR BOARD)) (NUMBERP X) (EQUAL X I)) (NOT (LESSP I I))). But this again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (AND (NOT (CAR BOARD)) (EQUAL (FIX X) I))) (NOT (LESSP X (ADD1 I))) (LESSP X I)) (NOT (LEGALP2 X BOARD I))), which simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (AND (NOT (CAR BOARD)) (EQUAL (FIX X) I))) (NOT (LEGALP2 X (CDR BOARD) (ADD1 I))) (LESSP X I)) (NOT (LEGALP2 X BOARD I))), which simplifies, opening up NLISTP, NOT, FIX, AND, LEGALP2, and LESSP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP2-FACT1 (PROVE-LEMMA SUB1-DIFFERENCE (REWRITE) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))) This conjecture simplifies, using linear arithmetic, to the following four new formulas: Case 4. (IMPLIES (NOT (LESSP Y X)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))). Appealing to the lemma SUB1-ELIM, we now replace X by (ADD1 Z) to eliminate (SUB1 X). We use the type restriction lemma noted when SUB1 was introduced to constrain the new variable. We must thus prove three new conjectures: Case 4.3. (IMPLIES (AND (EQUAL X 0) (NOT (LESSP Y X))) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which further simplifies, expanding the definitions of EQUAL, LESSP, DIFFERENCE, and SUB1, to: T. Case 4.2. (IMPLIES (AND (NOT (NUMBERP X)) (NOT (LESSP Y X))) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which further simplifies, appealing to the lemma SUB1-NNUMBERP, and opening up LESSP, DIFFERENCE, SUB1, and EQUAL, to: T. Case 4.1. (IMPLIES (AND (NUMBERP Z) (NOT (EQUAL (ADD1 Z) 0)) (NOT (LESSP Y (ADD1 Z)))) (EQUAL (SUB1 (DIFFERENCE (ADD1 Z) Y)) (DIFFERENCE Z Y))), which further simplifies, applying the lemma SUB1-ADD1, and opening up LESSP and DIFFERENCE, to: (IMPLIES (AND (NUMBERP Z) (NOT (EQUAL Y 0)) (NUMBERP Y) (NOT (LESSP (SUB1 Y) Z))) (EQUAL (SUB1 (DIFFERENCE Z (SUB1 Y))) (DIFFERENCE Z Y))). Appealing to the lemma SUB1-ELIM, we now replace Y by (ADD1 V) to eliminate (SUB1 Y). We use the type restriction lemma noted when SUB1 was introduced to constrain the new variable. We must thus prove: (IMPLIES (AND (NUMBERP V) (NUMBERP Z) (NOT (EQUAL (ADD1 V) 0)) (NOT (LESSP V Z))) (EQUAL (SUB1 (DIFFERENCE Z V)) (DIFFERENCE Z (ADD1 V)))). This further simplifies, applying SUB1-ADD1, and opening up the definition of DIFFERENCE, to the following two new formulas: Case 4.1.2. (IMPLIES (AND (NUMBERP V) (NUMBERP Z) (NOT (LESSP V Z)) (NOT (EQUAL Z 0))) (EQUAL (SUB1 (DIFFERENCE Z V)) (DIFFERENCE (SUB1 Z) V))), which we would usually push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us consider: (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y)), which we named *1 above. We will appeal to induction. Three inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP X) (p X Y)) (IMPLIES (AND (NOT (ZEROP X)) (ZEROP Y)) (p X Y)) (IMPLIES (AND (NOT (ZEROP X)) (NOT (ZEROP Y)) (p (SUB1 X) (SUB1 Y))) (p X Y))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform us that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for Y. The above induction scheme leads to the following three new goals: Case 3. (IMPLIES (ZEROP X) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))). This simplifies, using linear arithmetic, to the following four new goals: Case 3.4. (IMPLIES (AND (NOT (LESSP Y X)) (ZEROP X)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))). But this again simplifies, rewriting with the lemma SUB1-NNUMBERP, and opening up the functions ZEROP, EQUAL, DIFFERENCE, and SUB1, to: T. Case 3.3. (IMPLIES (AND (LESSP X Y) (ZEROP X)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, rewriting with the lemma SUB1-NNUMBERP, and opening up the definitions of ZEROP, EQUAL, DIFFERENCE, and SUB1, to: T. Case 3.2. (IMPLIES (AND (LESSP (SUB1 X) Y) (ZEROP X)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, applying the lemma SUB1-NNUMBERP, and expanding the functions ZEROP, EQUAL, DIFFERENCE, and SUB1, to: T. Case 3.1. (IMPLIES (AND (LESSP X 1) (ZEROP X)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, rewriting with SUB1-NNUMBERP, and unfolding the definitions of ZEROP, EQUAL, DIFFERENCE, and SUB1, to: T. Case 2. (IMPLIES (AND (NOT (ZEROP X)) (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))). This simplifies, using linear arithmetic, to the following four new goals: Case 2.4. (IMPLIES (AND (NOT (LESSP Y X)) (NOT (ZEROP X)) (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))). But this again simplifies, expanding the functions ZEROP, EQUAL, and DIFFERENCE, to: T. Case 2.3. (IMPLIES (AND (LESSP X Y) (NOT (ZEROP X)) (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, expanding the functions ZEROP, EQUAL, and DIFFERENCE, to: T. Case 2.2. (IMPLIES (AND (LESSP (SUB1 X) Y) (NOT (ZEROP X)) (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, unfolding ZEROP, EQUAL, and DIFFERENCE, to: T. Case 2.1. (IMPLIES (AND (LESSP X 1) (NOT (ZEROP X)) (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, expanding the definitions of ZEROP, EQUAL, and DIFFERENCE, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP X)) (NOT (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)))) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which simplifies, using linear arithmetic, to four new formulas: Case 1.4. (IMPLIES (AND (NOT (LESSP Y X)) (NOT (ZEROP X)) (NOT (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)))) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, opening up LESSP, ZEROP, and DIFFERENCE, to: (IMPLIES (AND (NOT (EQUAL Y 0)) (NUMBERP Y) (NOT (LESSP (SUB1 Y) (SUB1 X))) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y))) (EQUAL (SUB1 X) 0)) (EQUAL (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)) 0)). This again simplifies, expanding the definitions of EQUAL, LESSP, DIFFERENCE, and SUB1, to: T. Case 1.3. (IMPLIES (AND (LESSP X Y) (NOT (ZEROP X)) (NOT (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)))) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, unfolding the definitions of LESSP, ZEROP, and DIFFERENCE, to: (IMPLIES (AND (NOT (EQUAL Y 0)) (NUMBERP Y) (LESSP (SUB1 X) (SUB1 Y)) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y))) (EQUAL (SUB1 X) 0)) (EQUAL (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)) 0)). But this again simplifies, unfolding the definitions of EQUAL, LESSP, DIFFERENCE, and SUB1, to: T. Case 1.2. (IMPLIES (AND (LESSP (SUB1 X) Y) (NOT (ZEROP X)) (NOT (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)))) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, expanding LESSP, ZEROP, DIFFERENCE, and EQUAL, to two new formulas: Case 1.2.2. (IMPLIES (AND (NOT (EQUAL Y 0)) (NUMBERP Y) (EQUAL (SUB1 X) 0) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)))) (EQUAL (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)) 0)), which again simplifies, unfolding the functions EQUAL, DIFFERENCE, and SUB1, to: T. Case 1.2.1. (IMPLIES (AND (NOT (EQUAL Y 0)) (NUMBERP Y) (LESSP (SUB1 (SUB1 X)) (SUB1 Y)) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y))) (EQUAL (SUB1 X) 0)) (EQUAL (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)) 0)), which again simplifies, expanding SUB1, EQUAL, LESSP, and DIFFERENCE, to: T. Case 1.1. (IMPLIES (AND (LESSP X 1) (NOT (ZEROP X)) (NOT (ZEROP Y)) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)))) (EQUAL (SUB1 (DIFFERENCE X Y)) (DIFFERENCE (SUB1 X) Y))), which again simplifies, opening up ZEROP and DIFFERENCE, to: (IMPLIES (AND (LESSP X 1) (NOT (EQUAL X 0)) (NUMBERP X) (NOT (EQUAL Y 0)) (NUMBERP Y) (EQUAL (SUB1 (DIFFERENCE (SUB1 X) (SUB1 Y))) (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y))) (EQUAL (SUB1 X) 0)) (EQUAL (DIFFERENCE (SUB1 (SUB1 X)) (SUB1 Y)) 0)). But this again simplifies, using linear arithmetic, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] SUB1-DIFFERENCE (PROVE-LEMMA LEGALP1-IS-LEGALP2-GEN NIL (IMPLIES (AND (NUMBERP I) (NOT (LESSP X I))) (EQUAL (LEGALP1 (DIFFERENCE X I) BOARD) (LEGALP2 X BOARD I)))) Call the conjecture *1. We will try to prove it by induction. The recursive terms in the conjecture suggest six inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (ZEROP (DIFFERENCE X I)) (p X I BOARD)) (IMPLIES (AND (NOT (ZEROP (DIFFERENCE X I))) (NLISTP BOARD)) (p X I BOARD)) (IMPLIES (AND (NOT (ZEROP (DIFFERENCE X I))) (NOT (NLISTP BOARD)) (p X (ADD1 I) (CDR BOARD))) (p X I BOARD))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definitions of NLISTP and ZEROP can be used to prove that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for I. The above induction scheme leads to the following four new goals: Case 4. (IMPLIES (AND (ZEROP (DIFFERENCE X I)) (NUMBERP I) (NOT (LESSP X I))) (EQUAL (LEGALP1 (DIFFERENCE X I) BOARD) (LEGALP2 X BOARD I))). This simplifies, opening up the definitions of ZEROP, EQUAL, LEGALP1, and LEGALP2, to the following four new goals: Case 4.4. (IMPLIES (AND (EQUAL (DIFFERENCE X I) 0) (NUMBERP I) (NOT (LESSP X I)) (NOT (CAR BOARD)) (NUMBERP X) (NOT (EQUAL X I))) (EQUAL T (LEGALP2 X (CDR BOARD) (ADD1 I)))). However this again simplifies, using linear arithmetic, to: T. Case 4.3. (IMPLIES (AND (EQUAL (DIFFERENCE X I) 0) (NUMBERP I) (NOT (LESSP X I)) (NOT (CAR BOARD)) (NOT (NUMBERP X)) (NOT (EQUAL 0 I))) (EQUAL T (LEGALP2 X (CDR BOARD) (ADD1 I)))), which again simplifies, unfolding DIFFERENCE, EQUAL, and LESSP, to: T. Case 4.2. (IMPLIES (AND (EQUAL (DIFFERENCE X I) 0) (NUMBERP I) (NOT (LESSP X I)) (LISTP BOARD) (CAR BOARD)) (EQUAL F (LEGALP2 X (CDR BOARD) (ADD1 I)))), which again simplifies, trivially, to: (IMPLIES (AND (EQUAL (DIFFERENCE X I) 0) (NUMBERP I) (NOT (LESSP X I)) (LISTP BOARD) (CAR BOARD)) (NOT (LEGALP2 X (CDR BOARD) (ADD1 I)))), which further simplifies, using linear arithmetic and applying LEGALP2-FACT1, to: T. Case 4.1. (IMPLIES (AND (EQUAL (DIFFERENCE X I) 0) (NUMBERP I) (NOT (LESSP X I)) (NOT (LISTP BOARD))) (CAR BOARD)). But this again simplifies, rewriting with CAR-NLISTP, to: (IMPLIES (AND (EQUAL (DIFFERENCE X I) 0) (NUMBERP I) (NOT (LESSP X I)) (NOT (LISTP BOARD))) 0), which again simplifies, obviously, to: T. Case 3. (IMPLIES (AND (NOT (ZEROP (DIFFERENCE X I))) (NLISTP BOARD) (NUMBERP I) (NOT (LESSP X I))) (EQUAL (LEGALP1 (DIFFERENCE X I) BOARD) (LEGALP2 X BOARD I))). This simplifies, using linear arithmetic, applying the lemma LEGALP1-FACT1, and expanding the functions ZEROP, NLISTP, LENGTH, LEGALP2, and EQUAL, to: T. Case 2. (IMPLIES (AND (NOT (ZEROP (DIFFERENCE X I))) (NOT (NLISTP BOARD)) (LESSP X (ADD1 I)) (NUMBERP I) (NOT (LESSP X I))) (EQUAL (LEGALP1 (DIFFERENCE X I) BOARD) (LEGALP2 X BOARD I))). This simplifies, using linear arithmetic, to the following two new conjectures: Case 2.2. (IMPLIES (AND (NOT (NUMBERP X)) (NOT (ZEROP (DIFFERENCE X I))) (NOT (NLISTP BOARD)) (LESSP X (ADD1 I)) (NUMBERP I) (NOT (LESSP X I))) (EQUAL (LEGALP1 (DIFFERENCE X I) BOARD) (LEGALP2 X BOARD I))). This again simplifies, opening up DIFFERENCE and ZEROP, to: T. Case 2.1. (IMPLIES (AND (NUMBERP X) (NOT (ZEROP (DIFFERENCE I I))) (NOT (NLISTP BOARD)) (LESSP I (ADD1 I)) (NUMBERP I) (NOT (LESSP I I))) (EQUAL (LEGALP1 (DIFFERENCE I I) BOARD) (LEGALP2 I BOARD I))), which again simplifies, using linear arithmetic, applying the lemmas SUB1-ADD1 and LEGALP2-FACT1, and unfolding the functions ZEROP, NLISTP, LESSP, NUMBERP, DIFFERENCE, EQUAL, LEGALP1, ADD1, and LEGALP2, to two new formulas: Case 2.1.2. (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (DIFFERENCE I I) 0)) (LISTP BOARD) (LESSP (SUB1 I) I) (NUMBERP I) (NOT (LESSP I I)) (NOT (CAR BOARD))) (EQUAL (LEGALP1 (DIFFERENCE I I) BOARD) T)), which again simplifies, using linear arithmetic, to: T. Case 2.1.1. (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (DIFFERENCE I I) 0)) (LISTP BOARD) (LESSP (SUB1 I) I) (NUMBERP I) (NOT (LESSP I I)) (CAR BOARD)) (EQUAL (LEGALP1 (DIFFERENCE I I) BOARD) F)), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP (DIFFERENCE X I))) (NOT (NLISTP BOARD)) (EQUAL (LEGALP1 (DIFFERENCE X (ADD1 I)) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I))) (NUMBERP I) (NOT (LESSP X I))) (EQUAL (LEGALP1 (DIFFERENCE X I) BOARD) (LEGALP2 X BOARD I))), which simplifies, rewriting with the lemma SUB1-DIFFERENCE, and unfolding the definitions of ZEROP, NLISTP, LEGALP1, and LEGALP2, to five new goals: Case 1.5. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X I) 0)) (LISTP BOARD) (EQUAL (LEGALP1 (DIFFERENCE X (ADD1 I)) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I))) (NUMBERP I) (NOT (LESSP X I)) (CAR BOARD)) (EQUAL (LEGALP1 (DIFFERENCE (SUB1 X) I) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I)))), which further simplifies, using linear arithmetic, rewriting with the lemmas SUB1-ADD1, SUB1-NNUMBERP, and LEGALP2-FACT1, and opening up the functions DIFFERENCE, LESSP, EQUAL, LEGALP1, ADD1, and SUB1, to two new conjectures: Case 1.5.2. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X I) 0)) (LISTP BOARD) (NOT (NUMBERP X)) (EQUAL (LEGALP1 0 (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I))) (NUMBERP I) (EQUAL I 0) (CAR BOARD)) (CADR BOARD)), which again simplifies, opening up the definitions of DIFFERENCE and EQUAL, to: T. Case 1.5.1. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X I) 0)) (LISTP BOARD) (EQUAL X 0) (EQUAL (LEGALP1 0 (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I))) (NUMBERP I) (EQUAL I 0) (CAR BOARD)) (CADR BOARD)), which again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP 0 0) (NOT (EQUAL (DIFFERENCE 0 0) 0)) (LISTP BOARD) (EQUAL (LEGALP1 0 (CDR BOARD)) (LEGALP2 0 (CDR BOARD) 1)) (NUMBERP 0) (CAR BOARD)) (CADR BOARD)). This again simplifies, using linear arithmetic, to: T. Case 1.4. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X I) 0)) (LISTP BOARD) (EQUAL (LEGALP1 (DIFFERENCE X (ADD1 I)) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I))) (NUMBERP I) (NOT (LESSP X I)) (NOT (NUMBERP X)) (NOT (EQUAL 0 I))) (EQUAL (LEGALP1 (DIFFERENCE (SUB1 X) I) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I)))), which again simplifies, opening up the definitions of DIFFERENCE and EQUAL, to: T. Case 1.3. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X I) 0)) (LISTP BOARD) (EQUAL (LEGALP1 (DIFFERENCE X (ADD1 I)) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I))) (NUMBERP I) (NOT (LESSP X I)) (NUMBERP X) (NOT (EQUAL X I))) (EQUAL (LEGALP1 (DIFFERENCE (SUB1 X) I) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I)))), which further simplifies, rewriting with SUB1-ADD1, and unfolding DIFFERENCE, EQUAL, LESSP, and NUMBERP, to: T. Case 1.2. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X I) 0)) (LISTP BOARD) (EQUAL (LEGALP1 (DIFFERENCE X (ADD1 I)) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I))) (NUMBERP I) (NOT (LESSP X I)) (NOT (CAR BOARD)) (NUMBERP X) (EQUAL X I)) (EQUAL (LEGALP1 (DIFFERENCE (SUB1 X) I) (CDR BOARD)) T)). But this again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X I) 0)) (LISTP BOARD) (EQUAL (LEGALP1 (DIFFERENCE X (ADD1 I)) (CDR BOARD)) (LEGALP2 X (CDR BOARD) (ADD1 I))) (NUMBERP I) (NOT (LESSP X I)) (NOT (CAR BOARD)) (NOT (NUMBERP X)) (EQUAL 0 I)) (EQUAL (LEGALP1 (DIFFERENCE (SUB1 X) I) (CDR BOARD)) T)), which again simplifies, unfolding the functions DIFFERENCE and EQUAL, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.0 ] LEGALP1-IS-LEGALP2-GEN (DISABLE SUB1-DIFFERENCE) [ 0.0 0.0 0.0 ] SUB1-DIFFERENCE-OFF (PROVE-LEMMA LEGALP1-IS-LEGALP2 (REWRITE) (EQUAL (LEGALP1 X BOARD) (LEGALP2 X BOARD '0)) ((USE (LEGALP1-IS-LEGALP2-GEN (I '0))))) This conjecture simplifies, using linear arithmetic, applying LEGALP2-FACT1, and expanding NUMBERP, EQUAL, LESSP, NOT, AND, DIFFERENCE, IMPLIES, LEGALP1, ADD1, and LEGALP2, to the following two new conjectures: Case 2. (IMPLIES (AND (NOT (NUMBERP X)) (EQUAL (LEGALP1 0 BOARD) (LEGALP2 X BOARD 0)) (NOT (LISTP BOARD))) (CAR BOARD)). This again simplifies, applying CAR-NLISTP and LEGALP1-FACT2, and unfolding the definitions of NTH, EQUAL, and LEGALP2, to: (IMPLIES (AND (NOT (NUMBERP X)) (NOT (LISTP BOARD))) 0), which again simplifies, clearly, to: T. Case 1. (IMPLIES (AND (EQUAL X 0) (EQUAL (LEGALP1 0 BOARD) (LEGALP2 X BOARD 0)) (NOT (LISTP BOARD))) (CAR BOARD)). However this again simplifies, rewriting with the lemmas CAR-NLISTP and LEGALP1-FACT2, and opening up NTH, EQUAL, and LEGALP2, to: (IMPLIES (NOT (LISTP BOARD)) 0). This again simplifies, obviously, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP1-IS-LEGALP2 (PROVE-LEMMA LEGALP2-OPENER-F (REWRITE) (IMPLIES (NUMBERP I) (EQUAL (LEGALP2 X (CONS F BOARD) I) (OR (IF (EQUAL I 0) (ZEROP X) (EQUAL X I)) (LEGALP2 X BOARD (ADD1 I)))))) This simplifies, using linear arithmetic, applying LEGALP2-FACT1, CDR-CONS, and CAR-CONS, and unfolding the functions LEGALP2, ZEROP, and OR, to the following five new conjectures: Case 5. (IMPLIES (AND (NUMBERP I) (EQUAL I 0) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL X I)) (EQUAL T (LEGALP2 X BOARD (ADD1 I)))). This again simplifies, obviously, to: T. Case 4. (IMPLIES (AND (NUMBERP I) (EQUAL I 0) (NOT (NUMBERP X))) (EQUAL (EQUAL 0 I) T)). But this again simplifies, opening up NUMBERP and EQUAL, to: T. Case 3. (IMPLIES (AND (NUMBERP I) (EQUAL X I) (NOT (NUMBERP X))) (EQUAL (EQUAL 0 I) T)), which again simplifies, clearly, to: T. Case 2. (IMPLIES (AND (NUMBERP I) (EQUAL I 0) (EQUAL X 0) (NUMBERP X) (NOT (EQUAL X I))) (EQUAL (LEGALP2 X BOARD (ADD1 I)) T)). This again simplifies, clearly, to: T. Case 1. (IMPLIES (AND (NUMBERP I) (NOT (EQUAL I 0)) (NOT (EQUAL X I)) (NOT (NUMBERP X))) (EQUAL (EQUAL 0 I) (LEGALP2 X BOARD (ADD1 I)))). But this again simplifies, using linear arithmetic, applying LEGALP2-FACT1, and expanding the definition of EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP2-OPENER-F (PROVE-LEMMA LEGALP2-OPENER-X (REWRITE) (EQUAL (LEGALP2 X (CONS 'X BOARD) I) (LEGALP2 X BOARD (ADD1 I)))) This simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up the definition of LEGALP2, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP2-OPENER-X (PROVE-LEMMA LEGALP2-OPENER-O (REWRITE) (EQUAL (LEGALP2 X (CONS 'O BOARD) I) (LEGALP2 X BOARD (ADD1 I)))) This simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up the definition of LEGALP2, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP2-OPENER-O (PROVE-LEMMA LEGALP2-OPENER-NIL (REWRITE) (EQUAL (LEGALP2 X 'NIL I) F)) This conjecture simplifies, opening up the functions LISTP, LEGALP2, and EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP2-OPENER-NIL (DISABLE LEGALP2) [ 0.0 0.0 0.0 ] LEGALP2-OFF (DISABLE LEGALP) [ 0.0 0.0 0.0 ] LEGALP-OFF (PROVE-LEMMA LEGALP-SIMPLIFIER (REWRITE) (EQUAL (LEGALP X (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8)) (LEGALP2 X (LIST S0 S1 S2 S3 S4 S5 S6 S7 S8) 0))) This formula simplifies, rewriting with the lemmas CDR-CONS, LEGALP1-IS-LEGALP2, and LEGALP-IS-LEGALP1, and opening up the functions EQUAL, ADD1, and LENGTH, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP-SIMPLIFIER (DISABLE LEGALP-IS-LEGALP1) [ 0.0 0.0 0.0 ] LEGALP-IS-LEGALP1-OFF (DISABLE LEGALP1-IS-LEGALP2) [ 0.0 0.0 0.0 ] LEGALP1-IS-LEGALP2-OFF (DISABLE MEMBER) [ 0.0 0.0 0.0 ] MEMBER-OFF (DISABLE MOVE) [ 0.0 0.0 0.0 ] MOVE-OFF (PROVE-LEMMA NOOP-MOVE-GEN NIL (IMPLIES (NOT (LESSP I (LENGTH BOARD))) (EQUAL (MOVE X I BOARD) BOARD)) ((ENABLE MOVE))) Give the conjecture the name *1. We will appeal to induction. There are four plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (p X I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p X I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p X (SUB1 I) (CDR BOARD))) (p X I BOARD))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme produces the following four new formulas: Case 4. (IMPLIES (AND (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (NOT (LESSP I (LENGTH BOARD)))) (EQUAL (MOVE X I BOARD) BOARD)). This simplifies, opening up LENGTH, NOT, OR, EQUAL, LESSP, and MOVE, to the following three new formulas: Case 4.3. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (LISTP BOARD) (NOT (NUMBERP I))) (EQUAL (CONS X (CDR BOARD)) BOARD)). This again simplifies, using linear arithmetic, to: T. Case 4.2. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (LISTP BOARD) (EQUAL I 0)) (EQUAL (CONS X (CDR BOARD)) BOARD)), which again simplifies, using linear arithmetic, to: T. Case 4.1. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I)) (EQUAL (CONS (CAR BOARD) (MOVE X (SUB1 I) (CDR BOARD))) BOARD)), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL I 0) (NOT (NUMBERP I))) (NOT (LESSP I (LENGTH BOARD)))) (EQUAL (MOVE X I BOARD) BOARD)), which simplifies, expanding LENGTH, NOT, OR, EQUAL, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (LESSP (SUB1 I) (LENGTH (CDR BOARD))) (NOT (LESSP I (LENGTH BOARD)))) (EQUAL (MOVE X I BOARD) BOARD)), which simplifies, applying SUB1-ADD1, and opening up LENGTH, NOT, OR, and LESSP, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (EQUAL (MOVE X (SUB1 I) (CDR BOARD)) (CDR BOARD)) (NOT (LESSP I (LENGTH BOARD)))) (EQUAL (MOVE X I BOARD) BOARD)). This simplifies, appealing to the lemmas SUB1-ADD1 and CONS-CAR-CDR, and unfolding the functions LENGTH, NOT, OR, LESSP, and MOVE, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] NOOP-MOVE-GEN (PROVE-LEMMA NOOP-MOVE (REWRITE) (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (NUMBERP I) (NOT (EQUAL I 0)) (NOT (EQUAL I 1)) (NOT (EQUAL I 2)) (NOT (EQUAL I 3)) (NOT (EQUAL I 4)) (NOT (EQUAL I 5)) (NOT (EQUAL I 6)) (NOT (EQUAL I 7)) (NOT (EQUAL I 8))) (EQUAL (MOVE X I BOARD) BOARD)) ((USE (NOOP-MOVE-GEN)))) This simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] NOOP-MOVE (PROVE-LEMMA MOVE-ZEROP (REWRITE) (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (NOT (NUMBERP I))) (EQUAL (MOVE X I BOARD) (CONS X (CDR BOARD)))) ((ENABLE MOVE))) This simplifies, unfolding the definitions of LENGTH and MOVE, to: T. Q.E.D. [ 0.0 0.0 0.0 ] MOVE-ZEROP (PROVE-LEMMA NOT-LEGAL-MOVEP-X-X (REWRITE) (NOT (LEGAL-MOVEP BOARD 'X BOARD))) Name the conjecture *1. Let us appeal to the induction principle. There is only one suggested induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (EQUAL (CAR BOARD) (CAR BOARD)) (p (CDR BOARD))) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (EQUAL (CAR BOARD) (CAR BOARD))) (AND (EQUAL (CAR BOARD) F) (EQUAL (CAR BOARD) 'X))) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (EQUAL (CAR BOARD) (CAR BOARD))) (NOT (AND (EQUAL (CAR BOARD) F) (EQUAL (CAR BOARD) 'X)))) (p BOARD))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to four new conjectures: Case 4. (IMPLIES (NLISTP BOARD) (NOT (LEGAL-MOVEP BOARD 'X BOARD))), which simplifies, expanding NLISTP and LEGAL-MOVEP, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (EQUAL (CAR BOARD) (CAR BOARD)) (NOT (LEGAL-MOVEP (CDR BOARD) 'X (CDR BOARD)))) (NOT (LEGAL-MOVEP BOARD 'X BOARD))), which simplifies, unfolding the definitions of NLISTP and LEGAL-MOVEP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (EQUAL (CAR BOARD) (CAR BOARD))) (AND (EQUAL (CAR BOARD) F) (EQUAL (CAR BOARD) 'X))) (NOT (LEGAL-MOVEP BOARD 'X BOARD))), which simplifies, obviously, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (EQUAL (CAR BOARD) (CAR BOARD))) (NOT (AND (EQUAL (CAR BOARD) F) (EQUAL (CAR BOARD) 'X)))) (NOT (LEGAL-MOVEP BOARD 'X BOARD))). This simplifies, obviously, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] NOT-LEGAL-MOVEP-X-X (PROVE-LEMMA NOT-LEGAL-MOVEP-IF-OCCUPIED (REWRITE) (IMPLIES (NTH I BOARD) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))) ((ENABLE MOVE))) Give the conjecture the name *1. Let us appeal to the induction principle. The recursive terms in the conjecture suggest four inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I)) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (p (CDR BOARD) (SUB1 I))) (p BOARD I))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP can be used to show that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme generates the following four new conjectures: Case 4. (IMPLIES (AND (NLISTP BOARD) (NTH I BOARD)) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))). This simplifies, appealing to the lemmas CAR-NLISTP and NOT-LEGAL-MOVEP-X-X, and expanding the functions NLISTP, NTH, and MOVE, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I) (NTH I BOARD)) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))). This simplifies, applying NOT-LEGAL-MOVEP-X-X, CDR-CONS, and CAR-CONS, and expanding the definitions of NLISTP, ZEROP, EQUAL, NTH, MOVE, and LEGAL-MOVEP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (NTH (SUB1 I) (CDR BOARD))) (NTH I BOARD)) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))), which simplifies, expanding the functions NLISTP, ZEROP, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (LEGAL-MOVEP (CDR BOARD) 'X (MOVE 'X (SUB1 I) (CDR BOARD)))) (NTH I BOARD)) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))), which simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up NLISTP, ZEROP, NTH, MOVE, and LEGAL-MOVEP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] NOT-LEGAL-MOVEP-IF-OCCUPIED (PROVE-LEMMA NOT-LEGAL-MOVEP-IF-TOO-BIG (REWRITE) (IMPLIES (NOT (LESSP I (LENGTH BOARD))) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))) ((ENABLE MOVE))) Call the conjecture *1. Let us appeal to the induction principle. There are five plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (p BOARD I)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p BOARD I)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p (CDR BOARD) (SUB1 I))) (p BOARD I))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT inform us that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme produces four new formulas: Case 4. (IMPLIES (AND (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (NOT (LESSP I (LENGTH BOARD)))) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))), which simplifies, applying the lemma NOT-LEGAL-MOVEP-X-X, and opening up LENGTH, NOT, OR, EQUAL, LESSP, and MOVE, to four new conjectures: Case 4.4. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (NOT (LISTP BOARD))) (NOT (LEGAL-MOVEP BOARD 'X BOARD))), which again simplifies, using linear arithmetic, to: T. Case 4.3. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (LISTP BOARD) (NOT (NUMBERP I))) (NOT (LEGAL-MOVEP BOARD 'X (CONS 'X (CDR BOARD))))), which again simplifies, using linear arithmetic, to: T. Case 4.2. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (LISTP BOARD) (EQUAL I 0)) (NOT (LEGAL-MOVEP BOARD 'X (CONS 'X (CDR BOARD))))), which again simplifies, using linear arithmetic, to: T. Case 4.1. (IMPLIES (AND (EQUAL (ADD1 (LENGTH (CDR BOARD))) 0) (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I)) (NOT (LEGAL-MOVEP BOARD 'X (CONS (CAR BOARD) (MOVE 'X (SUB1 I) (CDR BOARD)))))), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL I 0) (NOT (NUMBERP I))) (NOT (LESSP I (LENGTH BOARD)))) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))), which simplifies, unfolding LENGTH, NOT, OR, EQUAL, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (LESSP (SUB1 I) (LENGTH (CDR BOARD))) (NOT (LESSP I (LENGTH BOARD)))) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))), which simplifies, rewriting with the lemma SUB1-ADD1, and opening up the definitions of LENGTH, NOT, OR, and LESSP, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (LEGAL-MOVEP (CDR BOARD) 'X (MOVE 'X (SUB1 I) (CDR BOARD)))) (NOT (LESSP I (LENGTH BOARD)))) (NOT (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)))), which simplifies, rewriting with the lemmas SUB1-ADD1, CDR-CONS, and CAR-CONS, and expanding the definitions of LENGTH, NOT, OR, LESSP, MOVE, and LEGAL-MOVEP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] NOT-LEGAL-MOVEP-IF-TOO-BIG (PROVE-LEMMA LEGAL-MOVEP-MOVE (REWRITE) (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (EQUAL (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)) (LEGALP I BOARD))) ((ENABLE LEGALP MOVE))) This conjecture simplifies, expanding LEGALP, to the following three new conjectures: Case 3. (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (NTH I BOARD)) (EQUAL (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)) F)). This again simplifies, rewriting with NOT-LEGAL-MOVEP-IF-OCCUPIED, and unfolding the definition of EQUAL, to: T. Case 2. (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (NOT (LESSP I 9))) (EQUAL (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)) F)). But this again simplifies, using linear arithmetic, applying NOT-LEGAL-MOVEP-IF-TOO-BIG, and expanding the function EQUAL, to: T. Case 1. (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (EQUAL (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD)) T)). This again simplifies, obviously, to the new conjecture: (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))), which we will name *1. We will appeal to induction. There are seven plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I)) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (p (CDR BOARD) (SUB1 I))) (p BOARD I))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme generates the following seven new conjectures: Case 7. (IMPLIES (AND (NLISTP BOARD) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))). This simplifies, unfolding the definitions of NLISTP, BOARDP, LENGTH, and EQUAL, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))). This simplifies, applying the lemmas ADD1-EQUAL, CDR-CONS, CAR-CONS, and MOVE-ZEROP, and unfolding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, NTH, MOVE, LEGAL-MOVEP, and ADD1, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (BOARDP (CDR BOARD))) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))). This simplifies, opening up NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (EQUAL (LENGTH (CDR BOARD)) 9)) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))). This simplifies, rewriting with ADD1-EQUAL, and unfolding NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and MOVE, to three new goals: Case 4.3. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (NOT (EQUAL (LENGTH (CDR BOARD)) 9)) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD)))) (LEGAL-MOVEP BOARD 'X (CONS 'O (MOVE 'X (SUB1 I) (CDR BOARD))))), which again simplifies, unfolding the function EQUAL, to: (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD)))) (LEGAL-MOVEP BOARD 'X (CONS 'O (MOVE 'X (SUB1 I) (CDR BOARD))))). Appealing to the lemma CAR-CDR-ELIM, we now replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). The result is: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL X 'O) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS 'O (MOVE 'X (SUB1 I) Z)))). But this further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 I) Z))). Appealing to the lemma SUB1-ELIM, we now replace I by (ADD1 X) to eliminate (SUB1 I). We employ the type restriction lemma noted when SUB1 was introduced to constrain the new variable. We must thus prove: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This further simplifies, obviously, to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1. Case 4.2. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (NOT (EQUAL (LENGTH (CDR BOARD)) 9)) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD)))) (LEGAL-MOVEP BOARD 'X (CONS 'X (MOVE 'X (SUB1 I) (CDR BOARD))))). But this again simplifies, unfolding EQUAL, to the formula: (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD)))) (LEGAL-MOVEP BOARD 'X (CONS 'X (MOVE 'X (SUB1 I) (CDR BOARD))))). Appealing to the lemma CAR-CDR-ELIM, we now replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). The result is: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL X 'X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS 'X (MOVE 'X (SUB1 I) Z)))). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 I) Z))). Applying the lemma SUB1-ELIM, replace I by (ADD1 X) to eliminate (SUB1 I). We use the type restriction lemma noted when SUB1 was introduced to restrict the new variable. This produces: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, clearly, to the new goal: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.2. Case 4.1. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (NOT (EQUAL (LENGTH (CDR BOARD)) 9)) (NOT (CAR BOARD)) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD)))) (LEGAL-MOVEP BOARD 'X (CONS F (MOVE 'X (SUB1 I) (CDR BOARD))))). But this again simplifies, unfolding the function EQUAL, to: (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (NOT (CAR BOARD)) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD)))) (LEGAL-MOVEP BOARD 'X (CONS F (MOVE 'X (SUB1 I) (CDR BOARD))))). Appealing to the lemma CAR-CDR-ELIM, we now replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). This generates: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS F (MOVE 'X (SUB1 I) Z)))). But this further simplifies, applying CDR-CONS and CAR-CONS, and opening up the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 I) Z))). Applying the lemma SUB1-ELIM, replace I by (ADD1 X) to eliminate (SUB1 I). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. This produces: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, trivially, to the new goal: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.3. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (LESSP (SUB1 I) 9)) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))). This simplifies, using linear arithmetic, to the new formula: (IMPLIES (AND (LESSP I 1) (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (LESSP (SUB1 I) 9)) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))), which again simplifies, using linear arithmetic, rewriting with ADD1-EQUAL, NOOP-MOVE, and NOT-LEGAL-MOVEP-X-X, and expanding the functions NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and ADD1, to the following three new goals: Case 3.3. (IMPLIES (AND (LESSP I 1) (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) 9)) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8)) (NTH (SUB1 I) (CDR BOARD))). This again simplifies, using linear arithmetic, to: T. Case 3.2. (IMPLIES (AND (LESSP I 1) (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) 9)) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8)) (NTH (SUB1 I) (CDR BOARD))), which again simplifies, using linear arithmetic, to: T. Case 3.1. (IMPLIES (AND (LESSP I 1) (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) 9)) (NOT (CAR BOARD)) (BOARDP (CDR BOARD)) (EQUAL (LENGTH (CDR BOARD)) 8) (LESSP (SUB1 I) 8)) (NTH (SUB1 I) (CDR BOARD))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NTH (SUB1 I) (CDR BOARD)) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))), which simplifies, applying the lemma ADD1-EQUAL, and unfolding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (LEGAL-MOVEP (CDR BOARD) 'X (MOVE 'X (SUB1 I) (CDR BOARD))) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (NTH I BOARD))) (LEGAL-MOVEP BOARD 'X (MOVE 'X I BOARD))), which simplifies, applying the lemmas ADD1-EQUAL, CDR-CONS, and CAR-CONS, and opening up the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, MOVE, and LEGAL-MOVEP, to: T. So next consider: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we named *1.3 above. This conjecture is subsumed by formula *1.2 above. So we now return to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we named *1.2 above. What luck! This conjecture is subsumed by formula *1.1 above. So next consider: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which is formula *1.1 above. We will try to prove it by induction. There are seven plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP Z) (p Z X)) (IMPLIES (AND (NOT (NLISTP Z)) (ZEROP X)) (p Z X)) (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (p (CDR Z) (SUB1 X))) (p Z X))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP establish that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for Z. The above induction scheme leads to seven new conjectures: Case 7. (IMPLIES (AND (NLISTP Z) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, unfolding the functions NLISTP, BOARDP, LENGTH, and EQUAL, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP Z)) (ZEROP X) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, rewriting with ADD1-EQUAL, CDR-CONS, and CAR-CONS, and expanding the definitions of NLISTP, ZEROP, NUMBERP, BOARDP, LENGTH, LESSP, EQUAL, NTH, MOVE, and LEGAL-MOVEP, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (BOARDP (CDR Z))) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, unfolding the functions NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (EQUAL (LENGTH (CDR Z)) 8)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, applying ADD1-EQUAL, and opening up NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and MOVE, to three new conjectures: Case 4.3. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 8)) (NUMBERP X) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (MOVE 'X (SUB1 X) (CDR Z))))), which again simplifies, unfolding the definition of EQUAL, to: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This generates the goal: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL V 'O) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS 'O (MOVE 'X (SUB1 X) W)))). But this further simplifies, applying CDR-CONS and CAR-CONS, and expanding EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Applying the lemma SUB1-ELIM, replace X by (ADD1 V) to eliminate (SUB1 X). We rely upon the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which further simplifies, clearly, to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.1. Case 4.2. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 8)) (NUMBERP X) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (MOVE 'X (SUB1 X) (CDR Z))))). This again simplifies, opening up the function EQUAL, to the formula: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This generates: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL V 'X) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS 'X (MOVE 'X (SUB1 X) W)))). This further simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Applying the lemma SUB1-ELIM, replace X by (ADD1 V) to eliminate (SUB1 X). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which further simplifies, trivially, to the new formula: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.2. Case 4.1. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 8)) (NUMBERP X) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (MOVE 'X (SUB1 X) (CDR Z))))). This again simplifies, expanding EQUAL, to the goal: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This generates the formula: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (NOT V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS F (MOVE 'X (SUB1 X) W)))). This further simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and expanding the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP (SUB1 X) 7) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Appealing to the lemma SUB1-ELIM, we now replace X by (ADD1 V) to eliminate (SUB1 X). We use the type restriction lemma noted when SUB1 was introduced to constrain the new variable. The result is the goal: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This further simplifies, trivially, to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.3. Case 3. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (LESSP (SUB1 X) 8)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, using linear arithmetic, to: (IMPLIES (AND (EQUAL X 0) (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (LESSP (SUB1 X) 8)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which again simplifies, unfolding the functions NLISTP and ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NTH (SUB1 X) (CDR Z)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, rewriting with ADD1-EQUAL, and unfolding NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (LEGAL-MOVEP (CDR Z) 'X (MOVE 'X (SUB1 X) (CDR Z))) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 8) (LESSP X 8) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, appealing to the lemmas ADD1-EQUAL, CDR-CONS, and CAR-CONS, and expanding NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, MOVE, and LEGAL-MOVEP, to: T. So next consider: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), named *1.1.3 above. What luck! This conjecture is subsumed by the subgoal we named *1.1.2 above. So we now return to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), named *1.1.2 above. You probably did not notice, but this conjecture is subsumed by another subgoal awaiting our attention, namely *1.1.1 above. So next consider: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which is formula *1.1.1 above. Perhaps we can prove it by induction. There are seven plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP W) (p W V)) (IMPLIES (AND (NOT (NLISTP W)) (ZEROP V)) (p W V)) (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (p (CDR W) (SUB1 V))) (p W V))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP establish that the measure (COUNT V) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for W. The above induction scheme leads to the following seven new goals: Case 7. (IMPLIES (AND (NLISTP W) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, opening up the functions NLISTP, BOARDP, LENGTH, and EQUAL, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP W)) (ZEROP V) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, appealing to the lemmas ADD1-EQUAL, CDR-CONS, and CAR-CONS, and expanding the definitions of NLISTP, ZEROP, NUMBERP, BOARDP, LENGTH, LESSP, EQUAL, NTH, MOVE, and LEGAL-MOVEP, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (BOARDP (CDR W))) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, expanding NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (EQUAL (LENGTH (CDR W)) 7)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, rewriting with ADD1-EQUAL, and unfolding the functions NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and MOVE, to three new conjectures: Case 4.3. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 7)) (NUMBERP V) (EQUAL (CAR W) 'O) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'O (MOVE 'X (SUB1 V) (CDR W))))), which again simplifies, unfolding the function EQUAL, to: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL (CAR W) 'O) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'O (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). The result is the goal: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL X 'O) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS 'O (MOVE 'X (SUB1 V) Z)))). But this further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and expanding the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Appealing to the lemma SUB1-ELIM, we now replace V by (ADD1 X) to eliminate (SUB1 V). We employ the type restriction lemma noted when SUB1 was introduced to constrain the new variable. The result is: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This further simplifies, obviously, to the new formula: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.1. Case 4.2. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 7)) (NUMBERP V) (EQUAL (CAR W) 'X) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'X (MOVE 'X (SUB1 V) (CDR W))))). This again simplifies, unfolding EQUAL, to the goal: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL (CAR W) 'X) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'X (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). The result is: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL X 'X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS 'X (MOVE 'X (SUB1 V) Z)))). This further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Applying the lemma SUB1-ELIM, replace V by (ADD1 X) to eliminate (SUB1 V). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We thus obtain: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, clearly, to the new conjecture: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.2. Case 4.1. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 7)) (NUMBERP V) (NOT (CAR W)) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS F (MOVE 'X (SUB1 V) (CDR W))))). This again simplifies, unfolding the function EQUAL, to: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (NOT (CAR W)) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS F (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). This generates the formula: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (NOT X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS F (MOVE 'X (SUB1 V) Z)))). This further simplifies, applying CDR-CONS and CAR-CONS, and opening up the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP (SUB1 V) 6) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Applying the lemma SUB1-ELIM, replace V by (ADD1 X) to eliminate (SUB1 V). We use the type restriction lemma noted when SUB1 was introduced to restrict the new variable. This produces: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, clearly, to the new goal: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.3. Case 3. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (LESSP (SUB1 V) 7)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, using linear arithmetic, to: (IMPLIES (AND (EQUAL V 0) (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (LESSP (SUB1 V) 7)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which again simplifies, unfolding NLISTP and ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NTH (SUB1 V) (CDR W)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which simplifies, applying the lemma ADD1-EQUAL, and opening up NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (LEGAL-MOVEP (CDR W) 'X (MOVE 'X (SUB1 V) (CDR W))) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 7) (LESSP V 7) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which simplifies, rewriting with ADD1-EQUAL, CDR-CONS, and CAR-CONS, and expanding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, MOVE, and LEGAL-MOVEP, to: T. So next consider: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), named *1.1.1.3 above. Ah ha! This conjecture is subsumed by formula *1.1.1.2 above. So let us turn our attention to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which is formula *1.1.1.2 above. But this conjecture is subsumed by formula *1.1.1.1 above. So let us turn our attention to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), named *1.1.1.1 above. Perhaps we can prove it by induction. The recursive terms in the conjecture suggest seven inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP Z) (p Z X)) (IMPLIES (AND (NOT (NLISTP Z)) (ZEROP X)) (p Z X)) (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (p (CDR Z) (SUB1 X))) (p Z X))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP establish that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for Z. The above induction scheme produces the following seven new formulas: Case 7. (IMPLIES (AND (NLISTP Z) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, expanding NLISTP, BOARDP, LENGTH, and EQUAL, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP Z)) (ZEROP X) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, rewriting with ADD1-EQUAL, CDR-CONS, and CAR-CONS, and opening up NLISTP, ZEROP, NUMBERP, BOARDP, LENGTH, LESSP, EQUAL, NTH, MOVE, and LEGAL-MOVEP, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (BOARDP (CDR Z))) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, unfolding the functions NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (EQUAL (LENGTH (CDR Z)) 6)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, rewriting with ADD1-EQUAL, and unfolding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and MOVE, to the following three new goals: Case 4.3. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 6)) (NUMBERP X) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (MOVE 'X (SUB1 X) (CDR Z))))). This again simplifies, opening up the definition of EQUAL, to the conjecture: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We must thus prove: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL V 'O) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS 'O (MOVE 'X (SUB1 X) W)))). But this further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Applying the lemma SUB1-ELIM, replace X by (ADD1 V) to eliminate (SUB1 X). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We thus obtain the new conjecture: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which further simplifies, trivially, to the new formula: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.1.1.1. Case 4.2. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 6)) (NUMBERP X) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (MOVE 'X (SUB1 X) (CDR Z))))). However this again simplifies, opening up EQUAL, to the goal: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This generates: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL V 'X) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS 'X (MOVE 'X (SUB1 X) W)))). This further simplifies, applying CDR-CONS and CAR-CONS, and expanding the definitions of EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Applying the lemma SUB1-ELIM, replace X by (ADD1 V) to eliminate (SUB1 X). We use the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which further simplifies, obviously, to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.1.1.2. Case 4.1. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 6)) (NUMBERP X) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (MOVE 'X (SUB1 X) (CDR Z))))). However this again simplifies, opening up the function EQUAL, to: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This generates: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (NOT V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS F (MOVE 'X (SUB1 X) W)))). But this further simplifies, applying CDR-CONS and CAR-CONS, and opening up the definitions of EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP (SUB1 X) 5) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Applying the lemma SUB1-ELIM, replace X by (ADD1 V) to eliminate (SUB1 X). We rely upon the type restriction lemma noted when SUB1 was introduced to restrict the new variable. This produces: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which further simplifies, obviously, to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.1.1.3. Case 3. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (LESSP (SUB1 X) 6)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, using linear arithmetic, to: (IMPLIES (AND (EQUAL X 0) (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (LESSP (SUB1 X) 6)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which again simplifies, expanding the definitions of NLISTP and ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NTH (SUB1 X) (CDR Z)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, rewriting with ADD1-EQUAL, and unfolding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (LEGAL-MOVEP (CDR Z) 'X (MOVE 'X (SUB1 X) (CDR Z))) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 6) (LESSP X 6) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, applying ADD1-EQUAL, CDR-CONS, and CAR-CONS, and opening up the functions NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, MOVE, and LEGAL-MOVEP, to: T. So let us turn our attention to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), named *1.1.1.1.3 above. Ah ha! This conjecture is subsumed by the subgoal we named *1.1.1.1.2 above. So we now return to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), named *1.1.1.1.2 above. But this conjecture is subsumed by the subgoal we named *1.1.1.1.1 above. So next consider: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), named *1.1.1.1.1 above. We will try to prove it by induction. There are seven plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP W) (p W V)) (IMPLIES (AND (NOT (NLISTP W)) (ZEROP V)) (p W V)) (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (p (CDR W) (SUB1 V))) (p W V))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP inform us that the measure (COUNT V) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for W. The above induction scheme leads to seven new goals: Case 7. (IMPLIES (AND (NLISTP W) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which simplifies, unfolding NLISTP, BOARDP, LENGTH, and EQUAL, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP W)) (ZEROP V) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which simplifies, applying ADD1-EQUAL, CDR-CONS, and CAR-CONS, and unfolding the definitions of NLISTP, ZEROP, NUMBERP, BOARDP, LENGTH, LESSP, EQUAL, NTH, MOVE, and LEGAL-MOVEP, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (BOARDP (CDR W))) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, opening up NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (EQUAL (LENGTH (CDR W)) 5)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, applying ADD1-EQUAL, and unfolding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and MOVE, to three new goals: Case 4.3. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 5)) (NUMBERP V) (EQUAL (CAR W) 'O) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'O (MOVE 'X (SUB1 V) (CDR W))))), which again simplifies, expanding EQUAL, to: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL (CAR W) 'O) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'O (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). This generates: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL X 'O) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS 'O (MOVE 'X (SUB1 V) Z)))). But this further simplifies, applying CDR-CONS and CAR-CONS, and unfolding the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Applying the lemma SUB1-ELIM, replace V by (ADD1 X) to eliminate (SUB1 V). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, obviously, to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.1.1.1. Case 4.2. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 5)) (NUMBERP V) (EQUAL (CAR W) 'X) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'X (MOVE 'X (SUB1 V) (CDR W))))). However this again simplifies, opening up EQUAL, to the conjecture: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL (CAR W) 'X) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'X (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). We must thus prove: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL X 'X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS 'X (MOVE 'X (SUB1 V) Z)))). But this further simplifies, applying CDR-CONS and CAR-CONS, and expanding the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Applying the lemma SUB1-ELIM, replace V by (ADD1 X) to eliminate (SUB1 V). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, trivially, to the new goal: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.1.1.2. Case 4.1. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 5)) (NUMBERP V) (NOT (CAR W)) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS F (MOVE 'X (SUB1 V) (CDR W))))). However this again simplifies, opening up EQUAL, to: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (NOT (CAR W)) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS F (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). The result is the goal: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (NOT X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS F (MOVE 'X (SUB1 V) Z)))). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up EQUAL and LEGAL-MOVEP, to the new conjecture: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP (SUB1 V) 4) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Applying the lemma SUB1-ELIM, replace V by (ADD1 X) to eliminate (SUB1 V). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We thus obtain: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, obviously, to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.1.1.3. Case 3. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (LESSP (SUB1 V) 5)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, using linear arithmetic, to the new conjecture: (IMPLIES (AND (EQUAL V 0) (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (LESSP (SUB1 V) 5)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which again simplifies, expanding NLISTP and ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NTH (SUB1 V) (CDR W)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which simplifies, appealing to the lemma ADD1-EQUAL, and opening up the functions NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (LEGAL-MOVEP (CDR W) 'X (MOVE 'X (SUB1 V) (CDR W))) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 5) (LESSP V 5) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which simplifies, applying ADD1-EQUAL, CDR-CONS, and CAR-CONS, and opening up the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, MOVE, and LEGAL-MOVEP, to: T. So we now return to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), named *1.1.1.1.1.3 above. But this conjecture is subsumed by the subgoal we named *1.1.1.1.1.2 above. So let us turn our attention to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which is formula *1.1.1.1.1.2 above. You probably did not notice, but this conjecture is subsumed by the subgoal we named *1.1.1.1.1.1 above. So next consider: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which is formula *1.1.1.1.1.1 above. Perhaps we can prove it by induction. There are seven plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP Z) (p Z X)) (IMPLIES (AND (NOT (NLISTP Z)) (ZEROP X)) (p Z X)) (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (p (CDR Z) (SUB1 X))) (p Z X))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP can be used to prove that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for Z. The above induction scheme produces seven new goals: Case 7. (IMPLIES (AND (NLISTP Z) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, expanding the functions NLISTP, BOARDP, LENGTH, and EQUAL, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP Z)) (ZEROP X) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, applying ADD1-EQUAL, CDR-CONS, and CAR-CONS, and opening up NLISTP, ZEROP, NUMBERP, BOARDP, LENGTH, LESSP, EQUAL, NTH, MOVE, and LEGAL-MOVEP, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (BOARDP (CDR Z))) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, expanding the functions NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (EQUAL (LENGTH (CDR Z)) 4)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, rewriting with the lemma ADD1-EQUAL, and expanding the functions NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and MOVE, to the following three new formulas: Case 4.3. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 4)) (NUMBERP X) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (MOVE 'X (SUB1 X) (CDR Z))))). But this again simplifies, unfolding the function EQUAL, to: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). The result is: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL V 'O) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS 'O (MOVE 'X (SUB1 X) W)))). But this further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Applying the lemma SUB1-ELIM, replace X by (ADD1 V) to eliminate (SUB1 X). We use the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove the new conjecture: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which further simplifies, obviously, to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.1.1.1.1.1. Case 4.2. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 4)) (NUMBERP X) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (MOVE 'X (SUB1 X) (CDR Z))))). However this again simplifies, unfolding the function EQUAL, to: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This generates the formula: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL V 'X) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS 'X (MOVE 'X (SUB1 X) W)))). This further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of EQUAL and LEGAL-MOVEP, to the new goal: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Applying the lemma SUB1-ELIM, replace X by (ADD1 V) to eliminate (SUB1 X). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We thus obtain the new conjecture: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which further simplifies, trivially, to the new goal: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.1.1.1.1.2. Case 4.1. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 4)) (NUMBERP X) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (MOVE 'X (SUB1 X) (CDR Z))))). However this again simplifies, expanding the definition of EQUAL, to: (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NUMBERP X) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (MOVE 'X (SUB1 X) (CDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). The result is: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (NOT V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP (CONS V W) 'X (CONS F (MOVE 'X (SUB1 X) W)))). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding EQUAL and LEGAL-MOVEP, to the new goal: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP (SUB1 X) 3) (NOT (NTH (SUB1 X) W))) (LEGAL-MOVEP W 'X (MOVE 'X (SUB1 X) W))). Applying the lemma SUB1-ELIM, replace X by (ADD1 V) to eliminate (SUB1 X). We employ the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We thus obtain: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which further simplifies, obviously, to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we will finally name *1.1.1.1.1.1.3. Case 3. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (LESSP (SUB1 X) 4)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, using linear arithmetic, to the new formula: (IMPLIES (AND (EQUAL X 0) (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (LESSP (SUB1 X) 4)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which again simplifies, expanding the functions NLISTP and ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NTH (SUB1 X) (CDR Z)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, rewriting with ADD1-EQUAL, and expanding NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (LEGAL-MOVEP (CDR Z) 'X (MOVE 'X (SUB1 X) (CDR Z))) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 4) (LESSP X 4) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, appealing to the lemmas ADD1-EQUAL, CDR-CONS, and CAR-CONS, and opening up the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, MOVE, and LEGAL-MOVEP, to: T. So let us turn our attention to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we named *1.1.1.1.1.1.3 above. Ah ha! This conjecture is subsumed by the subgoal we named *1.1.1.1.1.1.2 above. So let us turn our attention to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which we named *1.1.1.1.1.1.2 above. This conjecture is subsumed by formula *1.1.1.1.1.1.1 above. So we now return to: (IMPLIES (AND (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), named *1.1.1.1.1.1.1 above. We will appeal to induction. There are seven plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP W) (p W V)) (IMPLIES (AND (NOT (NLISTP W)) (ZEROP V)) (p W V)) (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (p (CDR W) (SUB1 V))) (p W V))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP inform us that the measure (COUNT V) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for W. The above induction scheme leads to the following seven new formulas: Case 7. (IMPLIES (AND (NLISTP W) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, expanding the definitions of NLISTP, BOARDP, LENGTH, and EQUAL, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP W)) (ZEROP V) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, appealing to the lemmas ADD1-EQUAL, CDR-CONS, and CAR-CONS, and opening up NLISTP, ZEROP, NUMBERP, BOARDP, LENGTH, LESSP, EQUAL, NTH, MOVE, and LEGAL-MOVEP, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (BOARDP (CDR W))) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, expanding the functions NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (EQUAL (LENGTH (CDR W)) 3)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, applying ADD1-EQUAL, and unfolding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and MOVE, to three new goals: Case 4.3. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 3)) (NUMBERP V) (EQUAL (CAR W) 'O) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'O (MOVE 'X (SUB1 V) (CDR W))))), which again simplifies, expanding the function EQUAL, to: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL (CAR W) 'O) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'O (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). This generates: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL X 'O) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS 'O (MOVE 'X (SUB1 V) Z)))). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Applying the lemma SUB1-ELIM, replace V by (ADD1 X) to eliminate (SUB1 V). We rely upon the type restriction lemma noted when SUB1 was introduced to restrict the new variable. This produces: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, clearly, to the new conjecture: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.1.1.1.1.1. Case 4.2. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 3)) (NUMBERP V) (EQUAL (CAR W) 'X) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'X (MOVE 'X (SUB1 V) (CDR W))))). This again simplifies, opening up the definition of EQUAL, to: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL (CAR W) 'X) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS 'X (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). The result is: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (EQUAL X 'X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS 'X (MOVE 'X (SUB1 V) Z)))). But this further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the definitions of EQUAL and LEGAL-MOVEP, to: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Applying the lemma SUB1-ELIM, replace V by (ADD1 X) to eliminate (SUB1 V). We rely upon the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, trivially, to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.1.1.1.1.2. Case 4.1. (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NOT (EQUAL (LENGTH (CDR W)) 3)) (NUMBERP V) (NOT (CAR W)) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS F (MOVE 'X (SUB1 V) (CDR W))))). But this again simplifies, expanding EQUAL, to the formula: (IMPLIES (AND (LISTP W) (NOT (EQUAL V 0)) (NUMBERP V) (NOT (CAR W)) (BOARDP (CDR W)) (EQUAL (LENGTH (CDR W)) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) (CDR W)))) (LEGAL-MOVEP W 'X (CONS F (MOVE 'X (SUB1 V) (CDR W))))). Appealing to the lemma CAR-CDR-ELIM, we now replace W by (CONS X Z) to eliminate (CAR W) and (CDR W). The result is: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (NOT X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP (CONS X Z) 'X (CONS F (MOVE 'X (SUB1 V) Z)))). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of EQUAL and LEGAL-MOVEP, to the new goal: (IMPLIES (AND (NOT (EQUAL V 0)) (NUMBERP V) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP (SUB1 V) 2) (NOT (NTH (SUB1 V) Z))) (LEGAL-MOVEP Z 'X (MOVE 'X (SUB1 V) Z))). Applying the lemma SUB1-ELIM, replace V by (ADD1 X) to eliminate (SUB1 V). We use the type restriction lemma noted when SUB1 was introduced to restrict the new variable. This produces: (IMPLIES (AND (NUMBERP X) (NOT (EQUAL (ADD1 X) 0)) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which further simplifies, clearly, to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we will finally name *1.1.1.1.1.1.1.3. Case 3. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (LESSP (SUB1 V) 3)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, using linear arithmetic, to: (IMPLIES (AND (EQUAL V 0) (NOT (NLISTP W)) (NOT (ZEROP V)) (NOT (LESSP (SUB1 V) 3)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which again simplifies, unfolding the functions NLISTP and ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (NTH (SUB1 V) (CDR W)) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))), which simplifies, rewriting with ADD1-EQUAL, and unfolding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP W)) (NOT (ZEROP V)) (LEGAL-MOVEP (CDR W) 'X (MOVE 'X (SUB1 V) (CDR W))) (NUMBERP V) (BOARDP W) (EQUAL (LENGTH W) 3) (LESSP V 3) (NOT (NTH V W))) (LEGAL-MOVEP W 'X (MOVE 'X V W))). This simplifies, rewriting with ADD1-EQUAL, CDR-CONS, and CAR-CONS, and expanding the definitions of NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, MOVE, and LEGAL-MOVEP, to: T. So let us turn our attention to: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we named *1.1.1.1.1.1.1.3 above. Ah ha! This conjecture is subsumed by another subgoal awaiting our attention, namely *1.1.1.1.1.1.1.2 above. So next consider: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which is formula *1.1.1.1.1.1.1.2 above. Ah ha! This conjecture is subsumed by another subgoal awaiting our attention, namely *1.1.1.1.1.1.1.1 above. So next consider: (IMPLIES (AND (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which we named *1.1.1.1.1.1.1.1 above. We will try to prove it by induction. The recursive terms in the conjecture suggest seven inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP Z) (p Z X)) (IMPLIES (AND (NOT (NLISTP Z)) (ZEROP X)) (p Z X)) (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (p (CDR Z) (SUB1 X))) (p Z X))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP can be used to prove that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for Z. The above induction scheme leads to the following seven new formulas: Case 7. (IMPLIES (AND (NLISTP Z) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, opening up the definitions of NLISTP, BOARDP, LENGTH, and EQUAL, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP Z)) (ZEROP X) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, applying the lemmas ADD1-EQUAL, CDR-CONS, and CAR-CONS, and opening up NLISTP, ZEROP, NUMBERP, BOARDP, LENGTH, LESSP, EQUAL, NTH, MOVE, and LEGAL-MOVEP, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (BOARDP (CDR Z))) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, unfolding the functions NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (EQUAL (LENGTH (CDR Z)) 2)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, rewriting with the lemma ADD1-EQUAL, and opening up the functions NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, and MOVE, to the following three new goals: Case 4.3. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 2)) (NUMBERP X) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (LESSP (SUB1 X) 1) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (MOVE 'X (SUB1 X) (CDR Z))))). But this again simplifies, using linear arithmetic, to the goal: (IMPLIES (AND (LISTP Z) (NOT (EQUAL 1 0)) (NOT (EQUAL 1 2)) (NUMBERP 1) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (LESSP (SUB1 1) 1) (NOT (NTH (SUB1 1) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (MOVE 'X (SUB1 1) (CDR Z))))). But this again simplifies, expanding EQUAL, NUMBERP, SUB1, LESSP, NTH, and MOVE, to two new formulas: Case 4.3.2. (IMPLIES (AND (LISTP Z) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (NOT (CADR Z)) (NOT (LISTP (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'O (CDR Z)))), which again simplifies, rewriting with CAR-NLISTP, to: T. Case 4.3.1. (IMPLIES (AND (LISTP Z) (EQUAL (CAR Z) 'O) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (NOT (CADR Z)) (LISTP (CDR Z))) (LEGAL-MOVEP Z 'X (CONS 'O (CONS 'X (CDDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z) and W by (CONS D C) to eliminate (CAR W) and (CDR W). The result is: (IMPLIES (AND (EQUAL V 'O) (BOARDP (CONS D C)) (EQUAL (LENGTH (CONS D C)) 1) (NOT D)) (LEGAL-MOVEP (CONS V (CONS D C)) 'X (CONS 'O (CONS 'X C)))). This further simplifies, rewriting with CDR-CONS, CAR-CONS, and ADD1-EQUAL, and unfolding the definitions of EQUAL, BOARDP, LENGTH, NUMBERP, and LEGAL-MOVEP, to: T. Case 4.2. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 2)) (NUMBERP X) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (LESSP (SUB1 X) 1) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (MOVE 'X (SUB1 X) (CDR Z))))). However this again simplifies, using linear arithmetic, to the goal: (IMPLIES (AND (LISTP Z) (NOT (EQUAL 1 0)) (NOT (EQUAL 1 2)) (NUMBERP 1) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (LESSP (SUB1 1) 1) (NOT (NTH (SUB1 1) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (MOVE 'X (SUB1 1) (CDR Z))))). This again simplifies, opening up the definitions of EQUAL, NUMBERP, SUB1, LESSP, NTH, and MOVE, to two new goals: Case 4.2.2. (IMPLIES (AND (LISTP Z) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (NOT (CADR Z)) (NOT (LISTP (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS 'X (CDR Z)))), which again simplifies, rewriting with CAR-NLISTP, to: T. Case 4.2.1. (IMPLIES (AND (LISTP Z) (EQUAL (CAR Z) 'X) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (NOT (CADR Z)) (LISTP (CDR Z))) (LEGAL-MOVEP Z 'X (CONS 'X (CONS 'X (CDDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z) and W by (CONS D C) to eliminate (CAR W) and (CDR W). The result is: (IMPLIES (AND (EQUAL V 'X) (BOARDP (CONS D C)) (EQUAL (LENGTH (CONS D C)) 1) (NOT D)) (LEGAL-MOVEP (CONS V (CONS D C)) 'X (CONS 'X (CONS 'X C)))). This further simplifies, rewriting with CDR-CONS, CAR-CONS, and ADD1-EQUAL, and unfolding EQUAL, BOARDP, LENGTH, NUMBERP, and LEGAL-MOVEP, to: T. Case 4.1. (IMPLIES (AND (LISTP Z) (NOT (EQUAL X 0)) (NOT (EQUAL (LENGTH (CDR Z)) 2)) (NUMBERP X) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (LESSP (SUB1 X) 1) (NOT (NTH (SUB1 X) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (MOVE 'X (SUB1 X) (CDR Z))))). But this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LISTP Z) (NOT (EQUAL 1 0)) (NOT (EQUAL 1 2)) (NUMBERP 1) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (LESSP (SUB1 1) 1) (NOT (NTH (SUB1 1) (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (MOVE 'X (SUB1 1) (CDR Z))))). This again simplifies, expanding the definitions of EQUAL, NUMBERP, SUB1, LESSP, NTH, and MOVE, to two new conjectures: Case 4.1.2. (IMPLIES (AND (LISTP Z) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (NOT (CADR Z)) (NOT (LISTP (CDR Z)))) (LEGAL-MOVEP Z 'X (CONS F (CDR Z)))), which again simplifies, rewriting with CAR-NLISTP, to: T. Case 4.1.1. (IMPLIES (AND (LISTP Z) (NOT (CAR Z)) (BOARDP (CDR Z)) (EQUAL (LENGTH (CDR Z)) 1) (NOT (CADR Z)) (LISTP (CDR Z))) (LEGAL-MOVEP Z 'X (CONS F (CONS 'X (CDDR Z))))). Appealing to the lemma CAR-CDR-ELIM, we now replace Z by (CONS V W) to eliminate (CAR Z) and (CDR Z) and W by (CONS D C) to eliminate (CAR W) and (CDR W). We must thus prove: (IMPLIES (AND (NOT V) (BOARDP (CONS D C)) (EQUAL (LENGTH (CONS D C)) 1) (NOT D)) (LEGAL-MOVEP (CONS V (CONS D C)) 'X (CONS F (CONS 'X C)))). But this further simplifies, applying CDR-CONS, CAR-CONS, and ADD1-EQUAL, and unfolding EQUAL, BOARDP, LENGTH, NUMBERP, and LEGAL-MOVEP, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (LESSP (SUB1 X) 2)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))). This simplifies, using linear arithmetic, to the new goal: (IMPLIES (AND (EQUAL X 0) (NOT (NLISTP Z)) (NOT (ZEROP X)) (NOT (LESSP (SUB1 X) 2)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which again simplifies, expanding NLISTP and ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (NTH (SUB1 X) (CDR Z)) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, rewriting with the lemma ADD1-EQUAL, and expanding NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP Z)) (NOT (ZEROP X)) (LEGAL-MOVEP (CDR Z) 'X (MOVE 'X (SUB1 X) (CDR Z))) (NUMBERP X) (BOARDP Z) (EQUAL (LENGTH Z) 2) (LESSP X 2) (NOT (NTH X Z))) (LEGAL-MOVEP Z 'X (MOVE 'X X Z))), which simplifies, applying ADD1-EQUAL, CDR-CONS, and CAR-CONS, and unfolding NLISTP, ZEROP, BOARDP, LENGTH, NUMBERP, LESSP, EQUAL, SUB1, NTH, MOVE, and LEGAL-MOVEP, to: T. That finishes the proof of *1.1.1.1.1.1.1.1, which, in turn, also finishes the proof of *1.1.1.1.1.1.1, which also finishes the proof of *1.1.1.1.1.1, which, in turn, also finishes the proof of *1.1.1.1.1, which, in turn, also finishes the proof of *1.1.1.1, which, in turn, also finishes the proof of *1.1.1, which, in turn, also finishes the proof of *1.1, which, consequently, also finishes the proof of *1. Q.E.D. [ 0.0 0.6 0.4 ] LEGAL-MOVEP-MOVE (DISABLE LEGAL-MOVEP) [ 0.0 0.0 0.0 ] LEGAL-MOVEP-OFF (PROVE-LEMMA TIC-TAC-TOE-CORRECT1 (REWRITE) (MEMBER (TIC-TAC-TOEP (TIC-TAC-TOE XMOVES)) '(DRAW O-WIN))) This formula can be simplified, using the abbreviation TIC-TAC-TOE, to the new conjecture: (MEMBER (TIC-TAC-TOEP (CONS (LIST F F F F F F F F F) (TIC-TAC-TOE1 XMOVES (LIST F F F F F F F F F)))) '(DRAW O-WIN)), which simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of CONS, EQUAL, ADD1, and TIC-TAC-TOEP, to 31 new conjectures: Case 31.(IMPLIES (AND (EQUAL (CAR XMOVES) 2) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 30.(IMPLIES (AND (EQUAL (CAR XMOVES) 4) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 29.(IMPLIES (AND (EQUAL (CAR XMOVES) 6) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 28.(IMPLIES (AND (EQUAL (CAR XMOVES) 8) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 27.(IMPLIES (AND (EQUAL (CAR XMOVES) 7) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 26.(IMPLIES (AND (EQUAL (CAR XMOVES) 5) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 25.(IMPLIES (AND (EQUAL (CAR XMOVES) 3) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 24.(IMPLIES (AND (EQUAL (CAR XMOVES) 1) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 23.(IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 22.(IMPLIES (AND (EQUAL (CAR XMOVES) 0) (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.(IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 25 new formulas: Case 21.25. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.24. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 21.23. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 21.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 21.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new conjectures: Case 21.17.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.17.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.17.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 21.17.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.17.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.17.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.17.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.17.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 21.17.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 21.17.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.17.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.17.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 21.17.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.17.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.17.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 21.17.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.17.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.17.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 21.17.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.17.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'X 'O 'X) 'X '((O X X X O O X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.17.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.17.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.17.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.17.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.17.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.17.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.17.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.17.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.17.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.17.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.17.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.17.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new conjectures: Case 21.16.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 21.16.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 21.16.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 21.16.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 21.16.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 21.16.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 21.16.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.16.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.16.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new formulas: Case 21.16.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 21.16.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 21.16.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 21.16.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.16.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.16.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.16.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 21.16.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.16.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 21.16.11.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.11.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.16.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 21.16.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.16.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.16.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 21.16.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.16.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.16.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.16.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 19 new goals: Case 21.15.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 21.15.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 21.15.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 21.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 21.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 21.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 21.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 21.15.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 21.15.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.15.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.15.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 21.15.13.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.13.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.15.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 21.15.13.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.13.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.15.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 21.15.13.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.13.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'X 'O 'X) 'X '((O X X X O O X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.15.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.15.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.15.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.15.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 21.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 21.14.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.14.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.14.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 21.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE and ENDP, to: T. Case 21.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 21.14.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.14.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.14.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.14.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 21.14.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.14.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 21.14.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.14.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 21.14.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.14.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 21.14.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE and ENDP, to: T. Case 21.14.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.14.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 21.14.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 21.14.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.14.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 21.14.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.14.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 21.14.8.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.8.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.14.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 21.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 21.13.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.13.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 21.13.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 21.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 21.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 21.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 21.13.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.13.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.13.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.13.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 21.13.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.13.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 21.13.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 21.13.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.13.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.13.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 21.13.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.13.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 21.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new conjectures: Case 21.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 21.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 21.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 21.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 21.12.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.12.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.12.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.12.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 21.12.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.12.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.12.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.12.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.12.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 21.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 21.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 21.11.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE and ENDP, to: T. Case 21.11.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.11.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 21.11.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 21.11.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.11.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.11.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.11.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.11.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 21.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 21.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 21.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 21.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 21.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 21.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 21.10.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 21.10.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 21.10.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 21.10.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 21.10.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 21.10.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.10.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.10.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.10.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 21.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 21.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 21.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (EQUAL (CADR XMOVES) 0)) (NUMBERP (CADR XMOVES)) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 7)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F 'X F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.(IMPLIES (AND (EQUAL (CAR XMOVES) 4) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 22 new formulas: Case 20.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 20.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 20.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 20.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 20.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 20.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 20.15.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.15.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.15.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 20.15.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'X F 'O 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'X F 'O 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 20.15.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.15.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.15.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 20.15.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 20.15.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X 'X 'O F) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'O 'X 'X F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'O 'X 'X F 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 20.15.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.15.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.15.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 20.15.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 20.15.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O F 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'X 'O F 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O F 'O 'X) 'X '((O X O X X O X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 20.15.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X F 'X 'O 'X) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 20.15.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X F 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X 'X F 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X F 'O 'X) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 20.15.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 20.15.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.15.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.15.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 20.15.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'X 'O 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O 'X 'O F) 'X '((O X O X X O X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 20.15.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X F 'X 'O 'X) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 20.15.8.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.8.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X 'X 'O F) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 20.15.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.15.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 20.15.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.15.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 20.15.7.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'X 'O 'O 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'X 'O 'O 'O F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.7.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'X 'O 'O 'O F) 'X '((O X X X X O O O X) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 20.15.7.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O F 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'X 'O F 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.7.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O F 'O 'X) 'X '((O X O X X O X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 20.15.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'X 'O 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O 'X 'O F) 'X '((O X O X X O X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.15.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'X F F 'O F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'X F F 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new conjectures: Case 20.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 20.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 20.14.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE and ENDP, to: T. Case 20.14.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.14.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 20.14.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 20.14.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'X 'O 'O 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'X 'O 'O 'O F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'X 'O 'O 'O F) 'X '((O X X X X O O O X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.14.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 20.14.11.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O F 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'X 'O F 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.11.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O F 'O 'X) 'X '((O X O X X O X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.14.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 20.14.11.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'X 'O 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.11.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O 'X 'O F) 'X '((O X O X X O X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.14.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'X 'O F 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F 'X 'X 'O F 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 20.14.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding MOVE and ENDP, to: T. Case 20.14.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.14.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 20.14.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'X 'O 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'X 'O 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 20.14.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.14.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.14.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.14.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 20.14.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'O 'X 'X 'X 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O F 'X) 'X '((O O X X X O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.14.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 20.14.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.14.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 20.14.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.14.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 20.14.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'O 'X 'X 'X 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O 'X F) 'X '((O O X X X O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.14.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F 'X 'X 'O F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O F 'X 'X 'O F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 20.14.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.14.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.14.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 20.14.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 20.14.7.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'X 'O 'O 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'X 'O 'O 'O F)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.7.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'X 'O 'O 'O F) 'X '((O X X X X O O O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.14.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 20.14.7.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'O 'X 'X 'X 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.7.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O F 'X) 'X '((O O X X X O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.14.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 20.14.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'O 'X 'X 'X 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O 'X F) 'X '((O O X X X O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.14.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'X 'O F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'X 'O F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 20.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE and ENDP, to: T. Case 20.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 20.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 20.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 20.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new formulas: Case 20.13.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.13.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 20.13.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.13.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 20.13.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'X 'O 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O 'X 'O F) 'X '((O X O X X O X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.13.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 20.13.11.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13.11.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X F 'X 'O 'X) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.13.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 20.13.11.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13.11.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X 'X 'O F) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.13.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.13.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.13.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.13.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F 'X 'O F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new conjectures: Case 20.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 20.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 20.12.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up MOVE and ENDP, to: T. Case 20.12.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 20.12.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.12.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 20.12.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O F 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'X 'O F 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'X 'O F 'O 'X) 'X '((O X O X X O X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.12.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 20.12.11.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12.11.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X F 'X 'O 'X) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.12.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 20.12.11.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X F 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X 'X F 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12.11.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X F 'O 'X) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.12.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.12.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.12.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.12.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'X F F 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'X F F 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))))) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O F 'X F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'O F 'X F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 20.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 20.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 20.11.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 20.11.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 20.11.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.11.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 20.11.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'O 'X 'X 'X 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O 'X F) 'X '((O O X X X O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.11.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.11.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.11.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.11.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X F 'X F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'O 'X F 'X F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))))) '(DRAW O-WIN))). However this again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O F F 'X F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'O F F 'X F F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 20.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding MOVE and ENDP, to: T. Case 20.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 20.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 20.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 20.10.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 20.10.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.10.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.10.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 20.10.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'O 'X 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'O 'X 'X 'X 'O F) 'X '((O X O O X X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.10.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.10.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.10.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.10.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'O 'X 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'O 'X 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'O 'X 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'O 'X 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 20.9.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 20.9.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 20.9.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 20.9.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 20.9.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.9.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new formulas: Case 20.9.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 20.9.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 20.9.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 20.9.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 20.9.7.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'X 'O 'O 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'X 'O 'O 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9.7.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'X 'O 'O 'O F) 'X '((O X X X X O O O X) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.9.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 20.9.7.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'O 'X 'X 'X 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9.7.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O F 'X) 'X '((O O X X X O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.9.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 20.9.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'O 'X 'X 'X 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'O 'X 'X 'X 'O 'O 'X F) 'X '((O O X X X O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 20.9.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.9.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'X 'O 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'X 'O 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X F 'X F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X F 'X F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 20.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 20.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 4) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 7)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F F 'X F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'O F F F 'X F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.(IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). However this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 25 new conjectures: Case 19.25. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.24. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.23. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 19.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 19.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 19.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding MOVE and ENDP, to: T. Case 19.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new conjectures: Case 19.17.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.17.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.17.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 19.17.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.17.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 19.17.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 19.17.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.17.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.17.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new goals: Case 19.17.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 19.17.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.17.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 19.17.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.17.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.17.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.17.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 19.17.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.17.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 19.17.11.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.11.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.17.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 19.17.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.17.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'O 'X F 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.17.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 19.17.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.17.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.17.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.17.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'O 'X F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'O 'X F 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 19 new formulas: Case 19.16.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 19.16.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.16.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.16.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 19.16.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 19.16.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.16.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 19.16.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE and ENDP, to: T. Case 19.16.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.16.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.16.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 19.16.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X F 'O) 'X '((X X O O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.16.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 19.16.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.16.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 19.16.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X 'X F 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.16.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))). This again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 19.16.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.16.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.16.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.16.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 19.16.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X F 'O) 'X '((X X O O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.16.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 19.16.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.16.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 19.16.8.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.8.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X 'X F 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.16.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.16.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 19.16.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.16.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new goals: Case 19.15.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.15.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 19.15.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 19.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 19.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 19.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 19.15.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE and ENDP, to: T. Case 19.15.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 19.15.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 19.15.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 19.15.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X F 'O 'O 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O 'X 'O 'X) 'X '((O X X X O O X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.15.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 19.15.10.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.10.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.15.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 19.15.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.15.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.15.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.15.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.15.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))). This again simplifies, applying the lemma MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 19.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new conjectures: Case 19.14.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE and ENDP, to: T. Case 19.14.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 19.14.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 19.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 19.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 19.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE and ENDP, to: T. Case 19.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 19.14.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.14.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 19.14.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 19.14.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 19.14.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.14.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))). However this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 19.14.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 19.14.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.14.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.14.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 19.14.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.14.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 19.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 19.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 19.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 19.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 19.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 19.13.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.13.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 19.13.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 19.13.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 19.13.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.13.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.13.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.13.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.13.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))). This again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))))) '(DRAW O-WIN))). This again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 19.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE and ENDP, to: T. Case 19.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 19.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 19.12.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE and ENDP, to: T. Case 19.12.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 19.12.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.12.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 19.12.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X F 'O 'O 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O 'X 'O 'X) 'X '((O X X X O O X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.12.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 19.12.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.12.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 19.12.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.12.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.12.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.12.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.12.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))))) '(DRAW O-WIN))). However this again simplifies, appealing to the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new formulas: Case 19.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 19.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 19.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new formulas: Case 19.11.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.11.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 19.11.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 19.11.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.11.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.11.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 19.11.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.11.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.11.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new formulas: Case 19.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 19.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 19.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 19.10.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 19.10.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 19.10.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 19.10.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.10.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.10.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 19.10.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 19.10.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.10.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 19.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 19.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 19.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (EQUAL (CADR XMOVES) 0)) (NUMBERP (CADR XMOVES)) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 7)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.(IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). However this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 25 new conjectures: Case 18.25. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up MOVE and ENDP, to: T. Case 18.24. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.23. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 18.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 18.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 18.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 19 new goals: Case 18.17.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 18.17.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 18.17.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 18.17.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.17.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.17.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE and ENDP, to: T. Case 18.17.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 18.17.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 18.17.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.17.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 18.17.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 18.17.13.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.13.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.17.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 18.17.13.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.13.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.17.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 18.17.13.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.13.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'X 'O 'X) 'X '((O X X X O O X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.17.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.17.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.17.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.17.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.17.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.17.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.17.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.17.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.17.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.17.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.17.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 18.16.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.16.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.16.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.16.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 18.16.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.16.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 18.16.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.16.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.16.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new formulas: Case 18.16.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 18.16.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 18.16.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.16.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 18.16.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.16.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 18.16.11.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.11.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.16.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.16.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.16.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.16.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))). This again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.16.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.16.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.16.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.16.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.16.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 18.15.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 18.15.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.15.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 18.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 18.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE and ENDP, to: T. Case 18.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new goals: Case 18.15.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 18.15.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 18.15.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.15.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 18.15.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 18.15.13.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.13.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.15.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.15.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.15.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.15.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.15.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 19 new formulas: Case 18.14.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 18.14.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 18.14.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 18.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 18.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 18.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 18.14.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 18.14.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 18.14.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 18.14.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 18.14.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X F 'O 'O 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O 'X 'O 'X) 'X '((O X X X O O X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.14.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 18.14.10.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.10.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.14.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 18.14.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.14.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.14.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.14.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.14.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))))) '(DRAW O-WIN))). But this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new conjectures: Case 18.13.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding MOVE and ENDP, to: T. Case 18.13.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.13.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 18.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new conjectures: Case 18.13.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.13.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 18.13.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.13.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.13.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 18.13.13.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.13.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.13.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 18.13.13.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.13.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.13.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 18.13.13.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.13.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O F 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.13.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 18.13.13.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.13.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O F 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.13.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.13.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.13.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.13.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.13.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))). This again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 19 new conjectures: Case 18.12.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 18.12.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 18.12.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 18.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 18.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new formulas: Case 18.12.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding MOVE and ENDP, to: T. Case 18.12.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.12.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 18.12.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.12.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 18.12.11.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.11.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.12.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 18.12.11.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.11.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.12.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 18.12.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X 'O 'X) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.12.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 18.12.11.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.11.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X 'O 'X) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.12.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.12.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.12.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.12.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.12.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))). This again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))))) '(DRAW O-WIN))). However this again simplifies, appealing to the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new conjectures: Case 18.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 18.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 18.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 18.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 18.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 18.11.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.11.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 18.11.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 18.11.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 18.11.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.11.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.11.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.11.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.11.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))))) '(DRAW O-WIN))). But this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 18.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 18.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 18.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 18.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 18.10.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 18.10.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 18.10.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 18.10.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 18.10.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 18.10.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.10.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.10.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.10.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 18.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 18.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 18.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (EQUAL (CADR XMOVES) 0)) (NUMBERP (CADR XMOVES)) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.(IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 25 new conjectures: Case 17.25. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 17.24. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.23. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 17.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 17.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 17.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 19 new formulas: Case 17.17.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 17.17.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 17.17.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 17.17.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 17.17.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.17.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up MOVE and ENDP, to: T. Case 17.17.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 17.17.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.17.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.17.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 17.17.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 17.17.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.17.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 17.17.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.17.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 17.17.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.17.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))). However this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 17.17.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.17.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.17.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.17.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 17.17.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.17.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 17.17.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.17.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 17.17.8.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.8.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.17.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.17.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.17.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.17.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O F F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O F F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new conjectures: Case 17.16.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 17.16.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 17.16.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.16.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 17.16.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.16.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.16.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 17.16.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.16.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 17.16.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.16.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 17.16.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.16.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 17.16.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.16.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 17.16.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'X 'O 'X 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'X 'O 'X 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'X 'O 'X 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.16.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 17.16.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 17.16.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.16.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.16.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 17.16.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.16.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 17.16.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.16.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 17.16.8.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'X 'O 'X 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'X 'O 'X 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.8.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'X 'O 'X 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.16.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.16.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.16.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.16.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new goals: Case 17.15.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 17.15.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.15.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 17.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new formulas: Case 17.15.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE and ENDP, to: T. Case 17.15.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.15.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.15.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 17.15.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 17.15.13.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.13.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.15.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.15.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.15.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.15.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.15.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))). This again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new goals: Case 17.14.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.14.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.14.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 17.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 17.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up MOVE and ENDP, to: T. Case 17.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 17.14.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up MOVE and ENDP, to: T. Case 17.14.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.14.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.14.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 17.14.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.14.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 17.14.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.14.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 17.14.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 17.14.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 17.14.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.14.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 19 new formulas: Case 17.13.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.13.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 17.13.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 17.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 17.13.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.13.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.13.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.13.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.13.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 17.13.13.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.13.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.13.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.13.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.13.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))). This again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))))) '(DRAW O-WIN))). However this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new goals: Case 17.12.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 17.12.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.12.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 17.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 17.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new conjectures: Case 17.12.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE and ENDP, to: T. Case 17.12.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 17.12.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 17.12.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding MOVE and ENDP, to: T. Case 17.12.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.12.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.12.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.12.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.12.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))). This again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 17.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 17.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 17.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 17.11.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 17.11.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 17.11.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.11.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 17.11.7.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11.7.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.11.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 17.11.7.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11.7.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.11.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 17.11.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'X 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.11.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.11.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 17.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 17.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 17.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 17.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 17.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 17.10.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 17.10.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 17.10.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 17.10.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 17.10.7.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10.7.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.10.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 17.10.7.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10.7.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.10.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 17.10.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'X 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 17.10.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.10.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 17.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 17.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 17.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (EQUAL (CADR XMOVES) 0)) (NUMBERP (CADR XMOVES)) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O F F 'X F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O F F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.(IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 25 new conjectures: Case 16.25. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE and ENDP, to: T. Case 16.24. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.23. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 16.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 16.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 16.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new goals: Case 16.17.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding MOVE and ENDP, to: T. Case 16.17.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.17.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.17.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.17.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 16.17.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 16.17.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 16.17.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.17.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 16.17.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.17.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.17.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.17.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))). However this again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new formulas: Case 16.17.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.17.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.17.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 16.17.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.17.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.17.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O 'X F F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.17.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 16.17.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.17.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X F 'O 'X F F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'X F 'O 'X F F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new conjectures: Case 16.16.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE and ENDP, to: T. Case 16.16.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.16.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 16.16.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 16.16.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 16.16.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding MOVE and ENDP, to: T. Case 16.16.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 16.16.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding MOVE and ENDP, to: T. Case 16.16.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 16.16.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 16.16.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.16.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X F 'O) 'X '((X X O O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.16.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 16.16.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.16.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 16.16.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X 'X F 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.16.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 16.16.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.16.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.16.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.16.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 16.16.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X F 'O) 'X '((X X O O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.16.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.16.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.16.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 16.16.8.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.8.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X 'X F 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.16.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F F 'O 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.16.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 16.16.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.16.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X 'X F 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X 'X F 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 16.15.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE and ENDP, to: T. Case 16.15.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 16.15.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 16.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 16.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 16.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 13 new formulas: Case 16.15.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 16.15.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 16.15.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 16.15.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.15.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.15.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 16.15.11.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.11.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.15.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 16.15.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.15.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.15.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 16.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))))) '(DRAW O-WIN))). This again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'O F 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F 'O F 'O 'X F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))))) '(DRAW O-WIN))). However this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new goals: Case 16.14.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE and ENDP, to: T. Case 16.14.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.14.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 16.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 16.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 16.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 16.14.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.14.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 16.14.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.14.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.14.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.14.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.14.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.14.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 16.14.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'X 'O 'X 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'X 'O 'X 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'X 'O 'X 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.14.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 16.14.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.14.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.14.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 16.14.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 16.14.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.14.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 16.14.8.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.8.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.14.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 16.14.8.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'X 'O 'X 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'X 'O 'X 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.8.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'X 'O 'X 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.14.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 16.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F F 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 16.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 16.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 16.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 16.13.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 16.13.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.13.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.13.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.13.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.13.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.13.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new goals: Case 16.12.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding MOVE and ENDP, to: T. Case 16.12.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 16.12.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 16.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 16.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 16.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 16.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 16.12.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 16.12.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 16.12.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.12.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.12.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.12.13.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.13.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.12.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.12.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.12.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))). This again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 16.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 16.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 16.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 16.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 16.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 16.11.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 16.11.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 16.11.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 16.11.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.11.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.11.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 16.11.10.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11.10.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'O 'X) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.11.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.11.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'X 'O) 'X '((X X O O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.11.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.11.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.11.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.11.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 16.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 16.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 16.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 16.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 16.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 16.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 16.10.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 16.10.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 16.10.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 16.10.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 16.10.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.10.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 16.10.10.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10.10.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'O 'X) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.10.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 16.10.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'X 'O) 'X '((X X O O O X X X O) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 16.10.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.10.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.10.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.10.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 16.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 16.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 16.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (EQUAL (CADR XMOVES) 0)) (NUMBERP (CADR XMOVES)) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 7)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F 'O 'X F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F F 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.(IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 25 new goals: Case 15.25. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.24. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.23. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 15.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 15.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 15.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 15.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 15.17.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 15.17.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 15.17.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 15.17.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.17.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.17.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.17.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.17.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 15.17.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 15.17.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 15.17.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 15.17.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 15.17.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.17.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.17.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 15.17.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.17.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.17.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 15.17.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.17.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'X 'O 'X) 'X '((O X X X O O X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.17.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.17.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.17.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.17.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.17.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.17.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.17.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.17.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.17.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.17.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.17.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.17.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F 'X 'X 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 15.16.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.16.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 15.16.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 15.16.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.16.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.16.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.16.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.16.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 15.16.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.16.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.16.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.16.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 15.16.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.16.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.16.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.16.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.16.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F F 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.16.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.16.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.16.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.16.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 19 new formulas: Case 15.15.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 15.15.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 15.15.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 15.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 15.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 15.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new conjectures: Case 15.15.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.15.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 15.15.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 15.15.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.15.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 15.15.13.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.13.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.15.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 15.15.13.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.13.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.15.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 15.15.13.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.13.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O F 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.15.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 15.15.13.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.13.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O F 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.15.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.15.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.15.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 15.15.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.15.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F F 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))). This again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 15.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 15.14.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 15.14.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.14.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 15.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 15.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 15.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 15.14.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.14.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 15.14.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.14.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.14.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 15.14.13.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O F 'X 'X 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.13.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O 'O 'O 'X 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.14.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.14.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.14.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'X 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'X 'X 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))). This again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 15.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F F F 'X 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 15.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 15.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 15.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 15.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 15.13.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 15.13.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 15.13.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 15.13.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.13.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.13.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.13.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F 'O 'X 'O 'X F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O F 'O 'X 'O 'X F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O F F 'X 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O F F 'X 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 15.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 15.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 15.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 15.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 15.12.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE and ENDP, to: T. Case 15.12.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 15.12.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.12.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.12.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.12.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 15.12.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.12.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.12.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 15.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 15.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 15.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 15.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 15.11.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 15.11.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 15.11.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 15.11.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.11.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.11.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 15.11.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.11.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 15.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 15.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 15.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 15.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 15.10.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 15.10.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 15.10.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 15.10.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.10.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.10.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 15.10.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 15.10.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (EQUAL (CADR XMOVES) 0)) (NUMBERP (CADR XMOVES)) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 7)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F F F 'X 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.(IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). This again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 25 new formulas: Case 14.25. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.24. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 14.23. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 14.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 14.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 14.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 14.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 14.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE and ENDP, to: T. Case 14.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new formulas: Case 14.17.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.17.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 14.17.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 14.17.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 14.17.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 14.17.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.17.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.17.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new formulas: Case 14.17.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 14.17.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 14.17.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 14.17.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.17.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.17.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 14.17.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.17.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.17.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.17.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.17.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.17.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O 'X 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.17.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.17.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.17.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.17.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.17.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.17.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.17.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.17.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F 'X 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F 'X 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 14.16.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.16.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.16.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 14.16.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.16.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 14.16.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.16.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.16.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 14.16.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.16.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.16.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 14.16.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 14.16.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X F 'O 'O 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O 'X 'O 'X) 'X '((O X X X O O X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.16.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 14.16.9.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16.9.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.16.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 14.16.9.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16.9.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.16.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.16.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.16.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.16.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X F F 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.16.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.16.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.16.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.16.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X F F 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X F F 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 14.15.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE and ENDP, to: T. Case 14.15.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.15.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 14.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 14.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 14.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 13 new conjectures: Case 14.15.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 14.15.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.15.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 14.15.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 14.15.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 14.15.11.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.11.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.15.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 14.15.11.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O 'X 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.11.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O 'X 'O F 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.15.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 14.15.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X 'O 'X) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.15.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 14.15.11.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (EQUAL (CADDDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.11.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (EQUAL (CADDDDR XMOVES) 5)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X 'O 'X) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.15.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.15.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.15.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying the lemma MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 14.15.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.15.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 14.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 14.14.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up MOVE and ENDP, to: T. Case 14.14.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 14.14.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 14.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 14.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 14.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 14.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 13 new formulas: Case 14.14.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up MOVE and ENDP, to: T. Case 14.14.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 14.14.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and unfolding the functions EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 14.14.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE and ENDP, to: T. Case 14.14.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.14.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.14.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and expanding the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 14.14.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.14.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (NUMBERP (CADDDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 0)) (NUMBERP (CADDDR XMOVES)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST F 'X 'O F 'O F 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))). But this again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 14.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))))) '(DRAW O-WIN))). This again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and opening up the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O F F 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O F F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, EQUAL, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 19 new formulas: Case 14.13.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.13.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 14.13.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 14.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, appealing to the lemma MOVE-ZEROP, and expanding EQUAL, LENGTH, CDR, CONS, and ENDP, to: T. Case 14.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 14.13.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 14.13.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 14.13.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 14.13.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.13.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 14.13.13.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.13.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X 'O 'X) 'X '((O X O X O X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.13.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.13.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.13.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'O F 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'O F 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))). However this again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and opening up the functions EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 14.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (NOT (NUMBERP (CADDR XMOVES))) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 0)) (NUMBERP (CADDR XMOVES)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST F 'X 'O F 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 14.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 14.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 14.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 14.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 14.12.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.12.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 14.12.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.12.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 14.12.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'O 'X 'X 'X 'O 'O 'O F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 7)) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X 'X 'O 'O 'O F 'X) 'X '((O X X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.12.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.12.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.12.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.12.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O 'O F F 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'O 'X 'X F 'O 'O F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'O 'X 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'O 'X 'X F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new conjectures: Case 14.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 14.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 14.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 14.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new formulas: Case 14.11.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding MOVE and ENDP, to: T. Case 14.11.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 14.11.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.11.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 14.11.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.11.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.11.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.11.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.11.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 14.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding MOVE and ENDP, to: T. Case 14.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 14.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 14.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 14.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 14.10.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 14.10.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 14.10.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 14.10.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 14.10.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 14.10.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.10.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.10.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.10.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (NUMBERP (CADR XMOVES))) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemma MOVE-ZEROP, and expanding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (EQUAL (CADR XMOVES) 0) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (EQUAL (CADR XMOVES) 0)) (NUMBERP (CADR XMOVES)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 7)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F 'X F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST F 'X F F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.(IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with the lemmas MOVE-ZEROP, LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, WINP, ADD1, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 22 new conjectures: Case 13.22. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 13.21. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 13.20. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 13.19. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 13.18. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 13.17. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 13.16. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.15. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 13.15.16. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 13.15.15. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 13.15.14. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 13.15.13. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 13.15.12. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 13.15.11. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.15.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 13.15.10.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 13.15.10.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.15.10.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 13.15.10.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 13.15.10.7.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.15.10.7.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.15.10.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.15.10.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.15.10.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.15.10.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.15.10.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.15.10.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.15.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.15.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.15.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.15.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.15.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.15.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.15.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.15.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.15.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new formulas: Case 13.14.16. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 13.14.15. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 13.14.14. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.14.13. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 13.14.12. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 13.14.11. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.14.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.14.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.14.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.14.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 13.14.7.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE and ENDP, to: T. Case 13.14.7.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.14.7.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 13.14.7.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.14.7.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.14.7.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 13.14.7.5.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14.7.5.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.14.7.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14.7.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14.7.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.14.7.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))). This again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.14.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.14.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new formulas: Case 13.13.16. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding MOVE and ENDP, to: T. Case 13.13.15. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 13.13.14. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.13.13. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 13.13.12. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 13.13.11. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.13.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new formulas: Case 13.13.10.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.13.10.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.13.10.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.13.10.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.13.10.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.13.10.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 13.13.10.5.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13.10.5.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.13.10.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13.10.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13.10.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.13.10.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.13.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.13.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.13.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.13.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.13.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new conjectures: Case 13.12.16. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding MOVE and ENDP, to: T. Case 13.12.15. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 13.12.14. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 13.12.13. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 13.12.12. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 13.12.11. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.12.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.12.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 13.12.9.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE and ENDP, to: T. Case 13.12.9.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.12.9.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.12.9.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 13.12.9.7.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12.9.7.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.12.9.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.12.9.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.12.9.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.12.9.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12.9.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12.9.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.12.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.12.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.12.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.12.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 13.11.16. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.11.15. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.11.14. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 13.11.13. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 13.11.12. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 13.11.11. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.11.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.11.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.11.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.11.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 13.11.7.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.11.7.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.11.7.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.11.7.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 13.11.7.7.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11.7.7.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.11.7.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 13.11.7.6.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11.7.6.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.11.7.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 13.11.7.5.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'X 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11.7.5.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.11.7.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.11.7.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.11.7.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.11.7.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.11.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.11.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new goals: Case 13.10.16. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 13.10.15. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 13.10.14. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 13.10.13. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.10.12. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 13.10.11. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.10.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new conjectures: Case 13.10.10.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.10.10.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.10.10.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 13.10.10.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 13.10.10.7.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10.10.7.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.10.10.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new goals: Case 13.10.10.6.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10.10.6.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'O 'X) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.10.10.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 13.10.10.5.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). However this again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10.10.5.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'X 'O) 'X '((X X O O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.10.10.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.10.10.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.10.10.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.10.10.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.10.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.10.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.10.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.10.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.10.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new formulas: Case 13.9.16. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 13.9.15. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 13.9.14. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 13.9.13. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.9.12. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 13.9.11. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.9.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.9.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.9.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 13.9.8.10. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 13.9.8.9. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 13.9.8.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 13.9.8.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 13.9.8.7.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9.8.7.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 13.9.8.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.9.8.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.9.8.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.9.8.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9.8.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9.8.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.9.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.9.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.9.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 13.8. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.7. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.6. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.5. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.4. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.2. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 13.1. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 7)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.(IMPLIES (AND (EQUAL (CAR XMOVES) 0) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))))) '(DRAW O-WIN))). However this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 22 new conjectures: Case 12.22. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 12.21. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 12.20. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.19. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 12.18. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 12.17. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 12.15.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, opening up MOVE and ENDP, to: T. Case 12.15.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 12.15.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 12.15.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 12.15.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 12.15.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.15.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 12.15.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE and ENDP, to: T. Case 12.15.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.15.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 12.15.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 12.15.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.15.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.15.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.15.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.15.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.15.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.15.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.15.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'X 'O 'O 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.15.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.15.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.15.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.15.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.15.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.15.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.15.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.15.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.15.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'X 'O F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new conjectures: Case 12.14.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 12.14.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 12.14.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 12.14.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 12.14.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 12.14.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.14.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.14.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.14.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.14.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new goals: Case 12.14.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, expanding MOVE and ENDP, to: T. Case 12.14.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 12.14.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 12.14.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.14.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.14.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new formulas: Case 12.14.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.14.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.14.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'X 'O F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.14.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.14.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'X 'O F 'O F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'X 'O F 'O F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). However this again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new conjectures: Case 12.13.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 12.13.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 12.13.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 12.13.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 12.13.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 12.13.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.13.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 12.13.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))). However this again simplifies, unfolding MOVE and ENDP, to: T. Case 12.13.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.13.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 12.13.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.13.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.13.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 12.13.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 2))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 2)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.13.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.13.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F 'O 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.13.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.13.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.13.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.13.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.13.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F 'O 'O F 'X F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F 'O 'O F 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 12.12.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 12.12.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 12.12.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 12.12.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.12.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 12.12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 12.12.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 12.12.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.12.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 12.12.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 12.12.9.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12.9.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). But this again simplifies, opening up the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.12.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.12.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.12.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.12.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 2)) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O F F 'O F 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))). This again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O F F 'O F F F 'X) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O F F 'O F F F 'X)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). This again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 12.11.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, expanding the functions MOVE and ENDP, to: T. Case 12.11.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 12.11.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 12.11.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 12.11.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 12.11.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.11.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.11.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.11.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.11.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following ten new conjectures: Case 12.11.7.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))). But this again simplifies, expanding MOVE and ENDP, to: T. Case 12.11.7.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.11.7.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE and ENDP, to: T. Case 12.11.7.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 12.11.7.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'X 'O 'O 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11.7.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'X 'O 'O 'O 'X F) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.11.7.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 12.11.7.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'O 'O 'X 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11.7.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'O 'O 'X 'X) 'X '((X O X X O O O X X) DRAW)) '(DRAW O-WIN))). However this again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.11.7.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new conjectures: Case 12.11.7.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 3))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X 'O 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X F 'O 'X 'O 'X 'O)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11.7.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 3)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O 'X 'O 'X 'O) 'X '((X O X X O X O X O) DRAW)) '(DRAW O-WIN))). This again simplifies, expanding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.11.7.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.11.7.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.11.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.11.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (EQUAL (CADDDR XMOVES) 3)) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X F 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.11.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (EQUAL (CADDR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.11.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 2)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F 'O 'X F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F F F 'O F 'O 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to 16 new conjectures: Case 12.10.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 12.10.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 12.10.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE and ENDP, to: T. Case 12.10.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE and ENDP, to: T. Case 12.10.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 12.10.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new formulas: Case 12.10.10.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 12.10.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.10.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 12.10.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to two new formulas: Case 12.10.10.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (EQUAL (CADDDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'X 'O 'O 'O 'X 'X 'O F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10.10.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (EQUAL (CADDDDR XMOVES) 8)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'X 'O 'O 'O 'X 'X 'O F) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, expanding TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.10.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 12.10.10.6.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'O 'X) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X 'O 'X)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10.10.6.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'O 'X) 'X '((X X O O O X X O X) DRAW)) '(DRAW O-WIN))), which again simplifies, unfolding the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.10.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))))) '(DRAW O-WIN))), which again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new goals: Case 12.10.10.5.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (EQUAL (CADDDDR XMOVES) 1))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). But this again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10.10.5.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (EQUAL (CADDDDR XMOVES) 1)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X 'X 'O) 'X '((X X O O O X X X O) DRAW)) '(DRAW O-WIN))). But this again simplifies, unfolding the definitions of TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.10.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.10.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.10.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (EQUAL (CADDDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.10.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (EQUAL (CADDDR XMOVES) 1)) (NOT (EQUAL (CADDDR XMOVES) 7)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O 'O 'O 'X 'X F F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X F 'O 'O 'O 'X 'X F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, appealing to the lemmas CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and opening up END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.10.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.10.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.10.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.10.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.10.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (NOT (EQUAL (CADDR XMOVES) 1)) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F 'O F 'O 'X F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X F 'O F 'O 'X F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (TIC-TAC-TOE1 (CDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, applying LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following 16 new goals: Case 12.9.16. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))). This again simplifies, unfolding MOVE and ENDP, to: T. Case 12.9.15. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding MOVE and ENDP, to: T. Case 12.9.14. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE and ENDP, to: T. Case 12.9.13. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE and ENDP, to: T. Case 12.9.12. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 12.9.11. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.9.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.9.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.9.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))), which again simplifies, applying the lemmas LEGALP2-OPENER-NIL, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and expanding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to ten new goals: Case 12.9.8.10. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE and ENDP, to: T. Case 12.9.8.9. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, opening up MOVE and ENDP, to: T. Case 12.9.8.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(DRAW))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE and ENDP, to: T. Case 12.9.8.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, rewriting with LEGALP2-OPENER-NIL, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP-SIMPLIFIER, TIC-TAC-TOE1-OPENER, CDR-CONS, and CAR-CONS, and opening up the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, ADD1, EQUAL, OTHER-PLAYER, LEGAL-MOVEP, BOARDP, and TIC-TAC-TOEP1, to the following two new conjectures: Case 12.9.8.7.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (EQUAL (CADDDDR XMOVES) 6))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X (CONS (MOVE 'X (CADDDDR XMOVES) (LIST 'X 'O 'X 'O 'O 'X F 'X 'O)) '(O-WIN))) '(DRAW O-WIN))). This again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-F, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9.8.7.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (EQUAL (CADDDDR XMOVES) 6)) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O 'X F 'X 'O) 'X '((X O X O O X X X O) DRAW)) '(DRAW O-WIN))). This again simplifies, opening up the functions TIC-TAC-TOEP1 and MEMBER, to: T. Case 12.9.8.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.9.8.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (TIC-TAC-TOE1 (CDDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.9.8.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.9.8.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9.8.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (EQUAL (CADDDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9.8.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (EQUAL (CADDDR XMOVES) 5)) (NOT (EQUAL (CADDDR XMOVES) 6)) (NOT (EQUAL (CADDDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X 'O 'O F F 'X F) 'X (CONS (MOVE 'X (CADDDR XMOVES) (LIST 'X 'O 'X 'O 'O F F 'X F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-O, LEGALP2-OPENER-X, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding the definitions of END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (TIC-TAC-TOE1 (CDDDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))))) '(DRAW O-WIN))). But this again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.9.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.9.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (EQUAL (CADDR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding MOVE, ENDP, PICK-MOVE, WINP, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.9.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (NOT (EQUAL (CADDR XMOVES) 3)) (NOT (EQUAL (CADDR XMOVES) 5)) (NOT (EQUAL (CADDR XMOVES) 6)) (NOT (EQUAL (CADDR XMOVES) 7)) (NOT (EQUAL (CADDR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X 'O 'X F 'O F F F F) 'X (CONS (MOVE 'X (CADDR XMOVES) (LIST 'X 'O 'X F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, rewriting with CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and expanding the functions END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 12.8. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 1) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.7. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 3) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.6. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 6) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.5. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 8) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.4. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 7) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.3. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 5) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (EQUAL (CADR XMOVES) 2) (NOT (ENDP (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 12.1. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (NOT (EQUAL (CADR XMOVES) 1)) (NOT (EQUAL (CADR XMOVES) 2)) (NOT (EQUAL (CADR XMOVES) 3)) (NOT (EQUAL (CADR XMOVES) 5)) (NOT (EQUAL (CADR XMOVES) 6)) (NOT (EQUAL (CADR XMOVES) 7)) (NOT (EQUAL (CADR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST 'X F F F 'O F F F F) 'X (CONS (MOVE 'X (CADR XMOVES) (LIST 'X F F F 'O F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying CDR-CONS, LEGAL-MOVEP-MOVE, LEGALP-SIMPLIFIER, LEGALP2-OPENER-X, LEGALP2-OPENER-O, LEGALP2-OPENER-F, LEGALP2-OPENER-NIL, BOARDP-MOVE, and CAR-CONS, and unfolding END, OTHER-PLAYER, ADD1, CONS, LENGTH, BOARDP, EQUAL, OR, TIC-TAC-TOEP1, and MEMBER, to: T. Case 11.(IMPLIES (AND (EQUAL (CAR XMOVES) 2) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))). But this again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 10.(IMPLIES (AND (EQUAL (CAR XMOVES) 4) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 9. (IMPLIES (AND (EQUAL (CAR XMOVES) 6) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 8. (IMPLIES (AND (EQUAL (CAR XMOVES) 8) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 7. (IMPLIES (AND (EQUAL (CAR XMOVES) 7) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 6. (IMPLIES (AND (EQUAL (CAR XMOVES) 5) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, expanding the functions MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 5. (IMPLIES (AND (EQUAL (CAR XMOVES) 3) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, unfolding the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 4. (IMPLIES (AND (EQUAL (CAR XMOVES) 1) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, opening up the definitions of MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 3. (IMPLIES (AND (NOT (NUMBERP (CAR XMOVES))) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))), which again simplifies, rewriting with MOVE-ZEROP, and unfolding the definitions of EQUAL, LENGTH, CDR, CONS, ENDP, PICK-MOVE, MOVE, and WINP, to: T. Case 2. (IMPLIES (AND (EQUAL (CAR XMOVES) 0) (NOT (ENDP (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F))) '(O-WIN)))) '(DRAW O-WIN))). However this again simplifies, opening up MOVE, ENDP, PICK-MOVE, and WINP, to: T. Case 1. (IMPLIES (AND (NOT (EQUAL (CAR XMOVES) 0)) (NUMBERP (CAR XMOVES)) (NOT (EQUAL (CAR XMOVES) 1)) (NOT (EQUAL (CAR XMOVES) 2)) (NOT (EQUAL (CAR XMOVES) 3)) (NOT (EQUAL (CAR XMOVES) 4)) (NOT (EQUAL (CAR XMOVES) 5)) (NOT (EQUAL (CAR XMOVES) 6)) (NOT (EQUAL (CAR XMOVES) 7)) (NOT (EQUAL (CAR XMOVES) 8))) (MEMBER (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS (MOVE 'X (CAR XMOVES) (LIST F F F F F F F F F)) '(O-WIN))) '(DRAW O-WIN))), which again simplifies, applying NOOP-MOVE, and unfolding the definitions of EQUAL, LENGTH, CONS, TIC-TAC-TOEP1, and MEMBER, to: T. Q.E.D. [ 0.0 8.6 14.3 ] TIC-TAC-TOE-CORRECT1 (DISABLE TIC-TAC-TOE1-OPENER) [ 0.0 0.0 0.0 ] TIC-TAC-TOE1-OPENER-OFF (DISABLE LEGALP-SIMPLIFIER) [ 0.0 0.0 0.0 ] LEGALP-SIMPLIFIER-OFF (DEFN X-MOVES-HONORED1 (BOARD XMOVES LST) (COND ((NLISTP LST) T) ((NLISTP (CDR LST)) T) (T (AND (EQUAL (CAR LST) (MOVE 'X (CAR XMOVES) BOARD)) (X-MOVES-HONORED1 (CADR LST) (CDR XMOVES) (CDDR LST)))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT LST) decreases according to the well-founded relation LESSP in each recursive call. Hence, X-MOVES-HONORED1 is accepted under the principle of definition. From the definition we can conclude that: (OR (FALSEP (X-MOVES-HONORED1 BOARD XMOVES LST)) (TRUEP (X-MOVES-HONORED1 BOARD XMOVES LST))) is a theorem. [ 0.0 0.0 0.0 ] X-MOVES-HONORED1 (DEFN X-MOVES-HONORED (XMOVES LST) (X-MOVES-HONORED1 (CAR LST) XMOVES (CDR LST))) From the definition we can conclude that: (OR (FALSEP (X-MOVES-HONORED XMOVES LST)) (TRUEP (X-MOVES-HONORED XMOVES LST))) is a theorem. [ 0.0 0.0 0.0 ] X-MOVES-HONORED (PROVE-LEMMA TIC-TAC-TOE-CORRECT2-LEMMA (REWRITE) (IMPLIES (BOARDP BOARD) (X-MOVES-HONORED1 BOARD XMOVES (TIC-TAC-TOE1 XMOVES BOARD))) ((ENABLE TIC-TAC-TOE1) (DISABLE LEGALP MOVE ENDP PICK-MOVE))) Give the conjecture the name *1. Let us appeal to the induction principle. Two inductions are suggested by terms in the conjecture, both of which are unflawed. So we will choose the one suggested by the largest number of nonprimitive recursive functions. We will induct according to the following scheme: (AND (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (p BOARD XMOVES)) (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (p BOARD XMOVES)) (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (p (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (CDR XMOVES))) (p BOARD XMOVES)) (IMPLIES (NOT (LEGALP (CAR XMOVES) BOARD)) (p BOARD XMOVES))). Linear arithmetic and the lemmas MOVE-NEVER-INCREASES-OPEN-SQUARES and LEGAL-MOVES-REDUCE-OPEN-SQUARES establish that the measure (OPEN-SQUARES BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for XMOVES. The above induction scheme generates five new conjectures: Case 5. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (ENDP (MOVE 'X (CAR XMOVES) BOARD)) (BOARDP BOARD)) (X-MOVES-HONORED1 BOARD XMOVES (TIC-TAC-TOE1 XMOVES BOARD))), which simplifies, applying CAR-CONS and CDR-CONS, and opening up TIC-TAC-TOE1, CDR, CAR, LISTP, and X-MOVES-HONORED1, to: T. Case 4. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (BOARDP BOARD)) (X-MOVES-HONORED1 BOARD XMOVES (TIC-TAC-TOE1 XMOVES BOARD))). This simplifies, applying the lemmas CAR-CONS and CDR-CONS, and unfolding the functions TIC-TAC-TOE1, X-MOVES-HONORED1, CDR, and LISTP, to: T. Case 3. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (NOT (BOARDP (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (BOARDP BOARD)) (X-MOVES-HONORED1 BOARD XMOVES (TIC-TAC-TOE1 XMOVES BOARD))). This simplifies, rewriting with BOARDP-MOVE, and opening up the functions EQUAL and OR, to: T. Case 2. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (X-MOVES-HONORED1 (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (CDR XMOVES) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (BOARDP BOARD)) (X-MOVES-HONORED1 BOARD XMOVES (TIC-TAC-TOE1 XMOVES BOARD))), which simplifies, applying CAR-CONS and CDR-CONS, and unfolding the definitions of TIC-TAC-TOE1 and X-MOVES-HONORED1, to: T. Case 1. (IMPLIES (AND (NOT (LEGALP (CAR XMOVES) BOARD)) (BOARDP BOARD)) (X-MOVES-HONORED1 BOARD XMOVES (TIC-TAC-TOE1 XMOVES BOARD))). This simplifies, applying CAR-CONS and CDR-CONS, and unfolding TIC-TAC-TOE1, CDR, CAR, LISTP, and X-MOVES-HONORED1, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.0 ] TIC-TAC-TOE-CORRECT2-LEMMA (PROVE-LEMMA TIC-TAC-TOE-CORRECT2 (REWRITE) (X-MOVES-HONORED XMOVES (TIC-TAC-TOE XMOVES))) WARNING: Note that the rewrite rule TIC-TAC-TOE-CORRECT2 will be stored so as to apply only to terms with the nonrecursive function symbol X-MOVES-HONORED. This formula can be simplified, using the abbreviation TIC-TAC-TOE, to the new formula: (X-MOVES-HONORED XMOVES (CONS (LIST F F F F F F F F F) (TIC-TAC-TOE1 XMOVES (LIST F F F F F F F F F)))), which simplifies, applying TIC-TAC-TOE-CORRECT2-LEMMA, CDR-CONS, and CAR-CONS, and opening up the definitions of BOARDP and X-MOVES-HONORED, to: T. Q.E.D. [ 0.0 0.0 0.0 ] TIC-TAC-TOE-CORRECT2 (DEFN INIT-GAME1 (N LST) (IF (ZEROP N) NIL (IF (NLISTP LST) NIL (IF (NLISTP (CDR LST)) (LIST (CAR LST)) (CONS (CAR LST) (CONS (CADR LST) (INIT-GAME1 (SUB1 N) (CDDR LST)))))))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of NLISTP and ZEROP inform us that the measure (COUNT N) decreases according to the well-founded relation LESSP in each recursive call. Hence, INIT-GAME1 is accepted under the definitional principle. The definition of INIT-GAME1 can be justified in another way. Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definitions of NLISTP and ZEROP establish that the measure (COUNT LST) decreases according to the well-founded relation LESSP in each recursive call. Observe that: (OR (LITATOM (INIT-GAME1 N LST)) (LISTP (INIT-GAME1 N LST))) is a theorem. [ 0.0 0.0 0.0 ] INIT-GAME1 (DEFN INIT-GAME (N LST) (CONS (CAR LST) (INIT-GAME1 N (CDR LST)))) Note that (LISTP (INIT-GAME N LST)) is a theorem. [ 0.0 0.0 0.0 ] INIT-GAME (PROVE-LEMMA TIC-TAC-TOE-CORRECT3-LEMMA (REWRITE) (EQUAL (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD))) T) ((INDUCT (TIC-TAC-TOE1 OPENING BOARD)) (ENABLE TIC-TAC-TOE1) (DISABLE LEGALP MOVE ENDP PICK-MOVE))) This formula can be simplified, using the abbreviations NOT, OR, and AND, to the following four new goals: Case 4. (IMPLIES (AND (LEGALP (CAR OPENING) BOARD) (ENDP (MOVE 'X (CAR OPENING) BOARD))) (EQUAL (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD))) T)). This simplifies, obviously, to the new formula: (IMPLIES (AND (LEGALP (CAR OPENING) BOARD) (ENDP (MOVE 'X (CAR OPENING) BOARD))) (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD)))). Applying the lemma CAR-CDR-ELIM, replace OPENING by (CONS X Z) to eliminate (CAR OPENING) and (CDR OPENING). This produces the following two new conjectures: Case 4.2. (IMPLIES (AND (NOT (LISTP OPENING)) (LEGALP (CAR OPENING) BOARD) (ENDP (MOVE 'X (CAR OPENING) BOARD))) (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD)))). But this further simplifies, rewriting with CAR-NLISTP, and opening up the functions LENGTH, APPEND, EQUAL, and INIT-GAME1, to: T. Case 4.1. (IMPLIES (AND (LEGALP X BOARD) (ENDP (MOVE 'X X BOARD))) (EQUAL (INIT-GAME1 (LENGTH (CONS X Z)) (TIC-TAC-TOE1 (APPEND (CONS X Z) XMOVES1) BOARD)) (INIT-GAME1 (LENGTH (CONS X Z)) (TIC-TAC-TOE1 (APPEND (CONS X Z) XMOVES2) BOARD)))). However this further simplifies, rewriting with CDR-CONS, CAR-CONS, and SUB1-ADD1, and unfolding the definitions of LENGTH, APPEND, TIC-TAC-TOE1, CDR, CAR, LISTP, INIT-GAME1, and CONS, to: T. Case 3. (IMPLIES (AND (LEGALP (CAR OPENING) BOARD) (NOT (ENDP (MOVE 'X (CAR OPENING) BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (EQUAL (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD))) T)). This simplifies, clearly, to the new formula: (IMPLIES (AND (LEGALP (CAR OPENING) BOARD) (NOT (ENDP (MOVE 'X (CAR OPENING) BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD)))). Applying the lemma CAR-CDR-ELIM, replace OPENING by (CONS X Z) to eliminate (CAR OPENING) and (CDR OPENING). This produces the following two new formulas: Case 3.2. (IMPLIES (AND (NOT (LISTP OPENING)) (LEGALP (CAR OPENING) BOARD) (NOT (ENDP (MOVE 'X (CAR OPENING) BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD)))). But this further simplifies, applying CAR-NLISTP, and expanding the definitions of LENGTH, APPEND, EQUAL, and INIT-GAME1, to: T. Case 3.1. (IMPLIES (AND (LEGALP X BOARD) (NOT (ENDP (MOVE 'X X BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)))) (EQUAL (INIT-GAME1 (LENGTH (CONS X Z)) (TIC-TAC-TOE1 (APPEND (CONS X Z) XMOVES1) BOARD)) (INIT-GAME1 (LENGTH (CONS X Z)) (TIC-TAC-TOE1 (APPEND (CONS X Z) XMOVES2) BOARD)))). This further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and SUB1-ADD1, and unfolding the definitions of LENGTH, APPEND, TIC-TAC-TOE1, INIT-GAME1, CONS, CAR, CDR, and LISTP, to: T. Case 2. (IMPLIES (AND (LEGALP (CAR OPENING) BOARD) (NOT (ENDP (MOVE 'X (CAR OPENING) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (EQUAL (EQUAL (INIT-GAME1 (LENGTH (CDR OPENING)) (TIC-TAC-TOE1 (APPEND (CDR OPENING) XMOVES1) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (INIT-GAME1 (LENGTH (CDR OPENING)) (TIC-TAC-TOE1 (APPEND (CDR OPENING) XMOVES2) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD))))) T)) (EQUAL (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD))) T)), which simplifies, unfolding LENGTH and APPEND, to two new goals: Case 2.2. (IMPLIES (AND (LEGALP (CAR OPENING) BOARD) (NOT (ENDP (MOVE 'X (CAR OPENING) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (EQUAL (INIT-GAME1 (LENGTH (CDR OPENING)) (TIC-TAC-TOE1 (APPEND (CDR OPENING) XMOVES1) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (INIT-GAME1 (LENGTH (CDR OPENING)) (TIC-TAC-TOE1 (APPEND (CDR OPENING) XMOVES2) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD))))) (NOT (LISTP OPENING))) (EQUAL (INIT-GAME1 0 (TIC-TAC-TOE1 XMOVES1 BOARD)) (INIT-GAME1 0 (TIC-TAC-TOE1 XMOVES2 BOARD)))), which again simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and expanding LENGTH, LISTP, APPEND, EQUAL, and INIT-GAME1, to: T. Case 2.1. (IMPLIES (AND (LEGALP (CAR OPENING) BOARD) (NOT (ENDP (MOVE 'X (CAR OPENING) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (EQUAL (INIT-GAME1 (LENGTH (CDR OPENING)) (TIC-TAC-TOE1 (APPEND (CDR OPENING) XMOVES1) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD)))) (INIT-GAME1 (LENGTH (CDR OPENING)) (TIC-TAC-TOE1 (APPEND (CDR OPENING) XMOVES2) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR OPENING) BOARD)) (MOVE 'X (CAR OPENING) BOARD))))) (LISTP OPENING)) (EQUAL (INIT-GAME1 (ADD1 (LENGTH (CDR OPENING))) (TIC-TAC-TOE1 (CONS (CAR OPENING) (APPEND (CDR OPENING) XMOVES1)) BOARD)) (INIT-GAME1 (ADD1 (LENGTH (CDR OPENING))) (TIC-TAC-TOE1 (CONS (CAR OPENING) (APPEND (CDR OPENING) XMOVES2)) BOARD)))). However this again simplifies, applying the lemmas CDR-CONS, CAR-CONS, and SUB1-ADD1, and expanding TIC-TAC-TOE1 and INIT-GAME1, to: T. Case 1. (IMPLIES (NOT (LEGALP (CAR OPENING) BOARD)) (EQUAL (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD))) T)), which simplifies, obviously, to: (IMPLIES (NOT (LEGALP (CAR OPENING) BOARD)) (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD)))). Applying the lemma CAR-CDR-ELIM, replace OPENING by (CONS X Z) to eliminate (CAR OPENING) and (CDR OPENING). We thus obtain the following two new conjectures: Case 1.2. (IMPLIES (AND (NOT (LISTP OPENING)) (NOT (LEGALP (CAR OPENING) BOARD))) (EQUAL (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) BOARD)) (INIT-GAME1 (LENGTH OPENING) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) BOARD)))). However this further simplifies, rewriting with CAR-NLISTP, and expanding LENGTH, APPEND, EQUAL, and INIT-GAME1, to: T. Case 1.1. (IMPLIES (NOT (LEGALP X BOARD)) (EQUAL (INIT-GAME1 (LENGTH (CONS X Z)) (TIC-TAC-TOE1 (APPEND (CONS X Z) XMOVES1) BOARD)) (INIT-GAME1 (LENGTH (CONS X Z)) (TIC-TAC-TOE1 (APPEND (CONS X Z) XMOVES2) BOARD)))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and SUB1-ADD1, and expanding the functions LENGTH, APPEND, TIC-TAC-TOE1, CDR, CAR, LISTP, INIT-GAME1, and CONS, to: T. Q.E.D. [ 0.0 0.3 0.1 ] TIC-TAC-TOE-CORRECT3-LEMMA (PROVE-LEMMA TIC-TAC-TOE-CORRECT3 (REWRITE) (EQUAL (EQUAL (INIT-GAME (LENGTH OPENING) (TIC-TAC-TOE (APPEND OPENING XMOVES1))) (INIT-GAME (LENGTH OPENING) (TIC-TAC-TOE (APPEND OPENING XMOVES2)))) T)) This conjecture can be simplified, using the abbreviation TIC-TAC-TOE, to: (EQUAL (EQUAL (INIT-GAME (LENGTH OPENING) (CONS (LIST F F F F F F F F F) (TIC-TAC-TOE1 (APPEND OPENING XMOVES1) (LIST F F F F F F F F F)))) (INIT-GAME (LENGTH OPENING) (CONS (LIST F F F F F F F F F) (TIC-TAC-TOE1 (APPEND OPENING XMOVES2) (LIST F F F F F F F F F))))) T). This simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and TIC-TAC-TOE-CORRECT3-LEMMA, and opening up the functions INIT-GAME and EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] TIC-TAC-TOE-CORRECT3 (PROVE-LEMMA TIC-TAC-TOE-CORRECT NIL (AND (MEMBER (TIC-TAC-TOEP (TIC-TAC-TOE XMOVES)) '(DRAW O-WIN)) (X-MOVES-HONORED XMOVES (TIC-TAC-TOE XMOVES)) (EQUAL (INIT-GAME (LENGTH OPENING) (TIC-TAC-TOE (APPEND OPENING XMOVES1))) (INIT-GAME (LENGTH OPENING) (TIC-TAC-TOE (APPEND OPENING XMOVES2))))) ((DISABLE MEMBER TIC-TAC-TOEP TIC-TAC-TOE X-MOVES-HONORED INIT-GAME))) This formula can be simplified, using the abbreviations AND and TIC-TAC-TOE-CORRECT3, to the following two new conjectures: Case 2. (MEMBER (TIC-TAC-TOEP (TIC-TAC-TOE XMOVES)) '(DRAW O-WIN)). This simplifies, rewriting with the lemma TIC-TAC-TOE-CORRECT1, to: T. Case 1. (X-MOVES-HONORED XMOVES (TIC-TAC-TOE XMOVES)). This simplifies, applying TIC-TAC-TOE-CORRECT2, to: T. Q.E.D. [ 0.0 0.0 0.0 ] TIC-TAC-TOE-CORRECT (PROVE-LEMMA TIC-TAC-TOEP1-OUTCOME (REWRITE) (IMPLIES (TIC-TAC-TOEP1 BOARD PLAYER LST) (EQUAL (TIC-TAC-TOEP1 BOARD PLAYER LST) (NTH (SUB1 (LENGTH LST)) LST)))) Call the conjecture *1. We will appeal to induction. There are three plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP LST) (p BOARD PLAYER LST)) (IMPLIES (AND (NOT (NLISTP LST)) (NOT (BOARDP (CAR LST)))) (p BOARD PLAYER LST)) (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (NOT (LEGAL-MOVEP BOARD PLAYER (CAR LST)))) (p BOARD PLAYER LST)) (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (WINP PLAYER (CAR LST))) (p BOARD PLAYER LST)) (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (ENDP (CAR LST))) (p BOARD PLAYER LST)) (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (NOT (ENDP (CAR LST))) (p (CAR LST) (OTHER-PLAYER PLAYER) (CDR LST))) (p BOARD PLAYER LST))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to establish that the measure (COUNT LST) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instances chosen for BOARD and PLAYER. The above induction scheme generates seven new formulas: Case 7. (IMPLIES (AND (NLISTP LST) (TIC-TAC-TOEP1 BOARD PLAYER LST)) (EQUAL (TIC-TAC-TOEP1 BOARD PLAYER LST) (NTH (SUB1 (LENGTH LST)) LST))), which simplifies, expanding the definitions of NLISTP, TIC-TAC-TOEP1, OTHER-PLAYER, UNPACK, EQUAL, and END, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP LST)) (NOT (BOARDP (CAR LST))) (TIC-TAC-TOEP1 BOARD PLAYER LST)) (EQUAL (TIC-TAC-TOEP1 BOARD PLAYER LST) (NTH (SUB1 (LENGTH LST)) LST))), which simplifies, opening up the definitions of NLISTP and TIC-TAC-TOEP1, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (NOT (LEGAL-MOVEP BOARD PLAYER (CAR LST))) (TIC-TAC-TOEP1 BOARD PLAYER LST)) (EQUAL (TIC-TAC-TOEP1 BOARD PLAYER LST) (NTH (SUB1 (LENGTH LST)) LST))), which simplifies, applying the lemma SUB1-ADD1, and expanding the definitions of NLISTP, TIC-TAC-TOEP1, OTHER-PLAYER, UNPACK, EQUAL, END, and LENGTH, to two new conjectures: Case 5.2. (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (NOT (LEGAL-MOVEP BOARD PLAYER (CAR LST))) (EQUAL PLAYER 'X) (EQUAL (CDR LST) '(O-WIN))) (EQUAL (TIC-TAC-TOEP1 BOARD 'X LST) (NTH (LENGTH (CDR LST)) LST))), which again simplifies, opening up END, OTHER-PLAYER, TIC-TAC-TOEP1, LENGTH, NTH, SUB1, NUMBERP, and EQUAL, to: T. Case 5.1. (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (NOT (LEGAL-MOVEP BOARD PLAYER (CAR LST))) (NOT (EQUAL PLAYER 'X)) (EQUAL (CDR LST) '(X-WIN))) (EQUAL 'X-WIN (NTH (LENGTH (CDR LST)) LST))), which again simplifies, opening up the definitions of LENGTH, NTH, SUB1, NUMBERP, and EQUAL, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (WINP PLAYER (CAR LST)) (TIC-TAC-TOEP1 BOARD PLAYER LST)) (EQUAL (TIC-TAC-TOEP1 BOARD PLAYER LST) (NTH (SUB1 (LENGTH LST)) LST))), which simplifies, rewriting with the lemma SUB1-ADD1, and expanding NLISTP, TIC-TAC-TOEP1, EQUAL, END, and LENGTH, to two new goals: Case 4.2. (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (WINP PLAYER (CAR LST)) (NOT (EQUAL PLAYER 'O)) (EQUAL (CDR LST) '(X-WIN))) (EQUAL 'X-WIN (NTH (LENGTH (CDR LST)) LST))), which again simplifies, expanding LENGTH, NTH, SUB1, NUMBERP, and EQUAL, to: T. Case 4.1. (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (WINP PLAYER (CAR LST)) (EQUAL PLAYER 'O) (EQUAL (CDR LST) '(O-WIN))) (EQUAL (TIC-TAC-TOEP1 BOARD 'O LST) (NTH (LENGTH (CDR LST)) LST))), which again simplifies, expanding the functions END, TIC-TAC-TOEP1, LENGTH, NTH, SUB1, NUMBERP, and EQUAL, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (ENDP (CAR LST)) (TIC-TAC-TOEP1 BOARD PLAYER LST)) (EQUAL (TIC-TAC-TOEP1 BOARD PLAYER LST) (NTH (SUB1 (LENGTH LST)) LST))), which simplifies, applying SUB1-ADD1, and opening up NLISTP, TIC-TAC-TOEP1, EQUAL, END, and LENGTH, to: (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (ENDP (CAR LST)) (EQUAL (CDR LST) '(DRAW))) (EQUAL 'DRAW (NTH (LENGTH (CDR LST)) LST))), which again simplifies, unfolding the functions LENGTH, NTH, SUB1, NUMBERP, and EQUAL, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (NOT (ENDP (CAR LST))) (NOT (TIC-TAC-TOEP1 (CAR LST) (OTHER-PLAYER PLAYER) (CDR LST))) (TIC-TAC-TOEP1 BOARD PLAYER LST)) (EQUAL (TIC-TAC-TOEP1 BOARD PLAYER LST) (NTH (SUB1 (LENGTH LST)) LST))), which simplifies, expanding the definitions of NLISTP and TIC-TAC-TOEP1, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP LST)) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (NOT (ENDP (CAR LST))) (EQUAL (TIC-TAC-TOEP1 (CAR LST) (OTHER-PLAYER PLAYER) (CDR LST)) (NTH (SUB1 (LENGTH (CDR LST))) (CDR LST))) (TIC-TAC-TOEP1 BOARD PLAYER LST)) (EQUAL (TIC-TAC-TOEP1 BOARD PLAYER LST) (NTH (SUB1 (LENGTH LST)) LST))), which simplifies, rewriting with SUB1-ADD1, and expanding NLISTP, TIC-TAC-TOEP1, LENGTH, and NTH, to: (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (NOT (ENDP (CAR LST))) (EQUAL (TIC-TAC-TOEP1 (CAR LST) (OTHER-PLAYER PLAYER) (CDR LST)) (NTH (SUB1 (LENGTH (CDR LST))) (CDR LST))) (NTH (SUB1 (LENGTH (CDR LST))) (CDR LST)) (EQUAL (LENGTH (CDR LST)) 0)) (EQUAL (NTH (SUB1 (LENGTH (CDR LST))) (CDR LST)) (CAR LST))), which again simplifies, opening up SUB1, EQUAL, and NTH, to: (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (NOT (ENDP (CAR LST))) (EQUAL (TIC-TAC-TOEP1 (CAR LST) (OTHER-PLAYER PLAYER) (CDR LST)) (CADR LST)) (CADR LST) (EQUAL (LENGTH (CDR LST)) 0)) (EQUAL (CADR LST) (CAR LST))). This further simplifies, opening up the function OTHER-PLAYER, to two new goals: Case 1.2. (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (NOT (ENDP (CAR LST))) (NOT (EQUAL PLAYER 'X)) (EQUAL (TIC-TAC-TOEP1 (CAR LST) 'X (CDR LST)) (CADR LST)) (CADR LST) (EQUAL (LENGTH (CDR LST)) 0)) (EQUAL (CADR LST) (CAR LST))). Applying the lemma CAR-CDR-ELIM, replace LST by (CONS X Z) to eliminate (CAR LST) and (CDR LST) and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This produces the following two new formulas: Case 1.2.2. (IMPLIES (AND (NOT (LISTP Z)) (BOARDP X) (LEGAL-MOVEP BOARD PLAYER X) (NOT (WINP PLAYER X)) (NOT (ENDP X)) (NOT (EQUAL PLAYER 'X)) (EQUAL (TIC-TAC-TOEP1 X 'X Z) (CAR Z)) (CAR Z) (EQUAL (LENGTH Z) 0)) (EQUAL (CAR Z) X)). However this further simplifies, rewriting with the lemma CAR-NLISTP, and expanding the definitions of END, EQUAL, OTHER-PLAYER, and TIC-TAC-TOEP1, to: T. Case 1.2.1. (IMPLIES (AND (BOARDP X) (LEGAL-MOVEP BOARD PLAYER X) (NOT (WINP PLAYER X)) (NOT (ENDP X)) (NOT (EQUAL PLAYER 'X)) (EQUAL (TIC-TAC-TOEP1 X 'X (CONS V W)) V) V (EQUAL (LENGTH (CONS V W)) 0)) (EQUAL V X)), which further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of OTHER-PLAYER, END, EQUAL, TIC-TAC-TOEP1, CONS, and LENGTH, to: T. Case 1.1. (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD PLAYER (CAR LST)) (NOT (WINP PLAYER (CAR LST))) (NOT (ENDP (CAR LST))) (EQUAL PLAYER 'X) (EQUAL (TIC-TAC-TOEP1 (CAR LST) 'O (CDR LST)) (CADR LST)) (CADR LST) (EQUAL (LENGTH (CDR LST)) 0)) (EQUAL (CADR LST) (CAR LST))). This again simplifies, obviously, to: (IMPLIES (AND (LISTP LST) (BOARDP (CAR LST)) (LEGAL-MOVEP BOARD 'X (CAR LST)) (NOT (WINP 'X (CAR LST))) (NOT (ENDP (CAR LST))) (EQUAL (TIC-TAC-TOEP1 (CAR LST) 'O (CDR LST)) (CADR LST)) (CADR LST) (EQUAL (LENGTH (CDR LST)) 0)) (EQUAL (CADR LST) (CAR LST))). Applying the lemma CAR-CDR-ELIM, replace LST by (CONS X Z) to eliminate (CAR LST) and (CDR LST) and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This produces the following two new conjectures: Case 1.1.2. (IMPLIES (AND (NOT (LISTP Z)) (BOARDP X) (LEGAL-MOVEP BOARD 'X X) (NOT (WINP 'X X)) (NOT (ENDP X)) (EQUAL (TIC-TAC-TOEP1 X 'O Z) (CAR Z)) (CAR Z) (EQUAL (LENGTH Z) 0)) (EQUAL (CAR Z) X)). However this finally simplifies, rewriting with CAR-NLISTP, and unfolding the functions END, EQUAL, OTHER-PLAYER, and TIC-TAC-TOEP1, to: T. Case 1.1.1. (IMPLIES (AND (BOARDP X) (LEGAL-MOVEP BOARD 'X X) (NOT (WINP 'X X)) (NOT (ENDP X)) (EQUAL (TIC-TAC-TOEP1 X 'O (CONS V W)) V) V (EQUAL (LENGTH (CONS V W)) 0)) (EQUAL V X)). But this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the definitions of OTHER-PLAYER, END, EQUAL, TIC-TAC-TOEP1, CONS, and LENGTH, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.1 ] TIC-TAC-TOEP1-OUTCOME (PROVE-LEMMA TIC-TAC-TOEP-OUTCOME (REWRITE) (IMPLIES (TIC-TAC-TOEP LST) (EQUAL (TIC-TAC-TOEP LST) (NTH (SUB1 (LENGTH LST)) LST)))) WARNING: Note that the rewrite rule TIC-TAC-TOEP-OUTCOME will be stored so as to apply only to terms with the nonrecursive function symbol TIC-TAC-TOEP. This formula can be simplified, using the abbreviations TIC-TAC-TOEP and IMPLIES, to the new conjecture: (IMPLIES (AND (LISTP LST) (EQUAL (CAR LST) (LIST F F F F F F F F F)) (TIC-TAC-TOEP1 (CAR LST) 'X (CDR LST))) (EQUAL (TIC-TAC-TOEP LST) (NTH (SUB1 (LENGTH LST)) LST))), which simplifies, rewriting with TIC-TAC-TOEP1-OUTCOME and SUB1-ADD1, and opening up EQUAL, TIC-TAC-TOEP, and LENGTH, to the new formula: (IMPLIES (AND (LISTP LST) (EQUAL (CAR LST) (LIST F F F F F F F F F)) (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CDR LST))) (EQUAL (NTH (SUB1 (LENGTH (CDR LST))) (CDR LST)) (NTH (LENGTH (CDR LST)) LST))), which again simplifies, expanding NTH, to: (IMPLIES (AND (LISTP LST) (EQUAL (CAR LST) (LIST F F F F F F F F F)) (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CDR LST)) (EQUAL (LENGTH (CDR LST)) 0)) (EQUAL (NTH (SUB1 (LENGTH (CDR LST))) (CDR LST)) (LIST F F F F F F F F F))). This again simplifies, unfolding the functions SUB1, EQUAL, and NTH, to: (IMPLIES (AND (LISTP LST) (EQUAL (CAR LST) (LIST F F F F F F F F F)) (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CDR LST)) (EQUAL (LENGTH (CDR LST)) 0)) (EQUAL (CADR LST) (LIST F F F F F F F F F))). Appealing to the lemma CAR-CDR-ELIM, we now replace LST by (CONS X Z) to eliminate (CAR LST) and (CDR LST) and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This generates two new goals: Case 2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL X (LIST F F F F F F F F F)) (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X Z) (EQUAL (LENGTH Z) 0)) (EQUAL (CAR Z) (LIST F F F F F F F F F))), which further simplifies, opening up the functions END, EQUAL, OTHER-PLAYER, and TIC-TAC-TOEP1, to: T. Case 1. (IMPLIES (AND (EQUAL X (LIST F F F F F F F F F)) (TIC-TAC-TOEP1 (LIST F F F F F F F F F) 'X (CONS V W)) (EQUAL (LENGTH (CONS V W)) 0)) (EQUAL V (LIST F F F F F F F F F))), which further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up the functions OTHER-PLAYER, END, EQUAL, TIC-TAC-TOEP1, ADD1, and LENGTH, to: T. Q.E.D. [ 0.0 0.0 0.0 ] TIC-TAC-TOEP-OUTCOME (PROVE-LEMMA PLUS-ADD1 (REWRITE) (EQUAL (PLUS X (ADD1 Y)) (ADD1 (PLUS X Y)))) This conjecture simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] PLUS-ADD1 (PROVE-LEMMA PLUS-COMMUTES1 (REWRITE) (EQUAL (PLUS X Y) (PLUS Y X))) WARNING: the newly proposed lemma, PLUS-COMMUTES1, could be applied whenever the previously added lemma PLUS-ADD1 could. This formula simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] PLUS-COMMUTES1 (PROVE-LEMMA PLUS-COMMUTES2 (REWRITE) (EQUAL (PLUS X Y Z) (PLUS Y X Z))) WARNING: the previously added lemma, PLUS-COMMUTES1, could be applied whenever the newly proposed PLUS-COMMUTES2 could! This simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] PLUS-COMMUTES2 (PROVE-LEMMA PLUS-ASSOCIATES (REWRITE) (EQUAL (PLUS (PLUS X Y) Z) (PLUS X Y Z))) WARNING: the previously added lemma, PLUS-COMMUTES1, could be applied whenever the newly proposed PLUS-ASSOCIATES could! This simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] PLUS-ASSOCIATES (PROVE-LEMMA TIMES-0 (REWRITE) (EQUAL (TIMES X 0) 0)) Call the conjecture *1. We will try to prove it by induction. There is only one suggested induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP X) (p X)) (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X))) (p X))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to establish that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates the following two new formulas: Case 2. (IMPLIES (ZEROP X) (EQUAL (TIMES X 0) 0)). This simplifies, expanding the definitions of ZEROP, TIMES, and EQUAL, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP X)) (EQUAL (TIMES (SUB1 X) 0) 0)) (EQUAL (TIMES X 0) 0)). This simplifies, opening up the functions ZEROP, TIMES, PLUS, and EQUAL, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] TIMES-0 (PROVE-LEMMA TIMES-NON-NUMBERP (REWRITE) (IMPLIES (NOT (NUMBERP Z)) (EQUAL (TIMES X Z) 0))) Call the conjecture *1. Perhaps we can prove it by induction. There is only one plausible induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP X) (p X Z)) (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Z)) (p X Z))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to prove that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to two new goals: Case 2. (IMPLIES (AND (ZEROP X) (NOT (NUMBERP Z))) (EQUAL (TIMES X Z) 0)), which simplifies, opening up the definitions of ZEROP, EQUAL, and TIMES, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP X)) (EQUAL (TIMES (SUB1 X) Z) 0) (NOT (NUMBERP Z))) (EQUAL (TIMES X Z) 0)), which simplifies, applying PLUS-COMMUTES1, and unfolding ZEROP, TIMES, EQUAL, and PLUS, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] TIMES-NON-NUMBERP (PROVE-LEMMA TIMES-ADD1 (REWRITE) (EQUAL (TIMES X (ADD1 Y)) (PLUS X (TIMES X Y)))) Give the conjecture the name *1. We will try to prove it by induction. There are three plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP X) (p X Y)) (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y)) (p X Y))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform us that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates two new goals: Case 2. (IMPLIES (ZEROP X) (EQUAL (TIMES X (ADD1 Y)) (PLUS X (TIMES X Y)))), which simplifies, applying PLUS-COMMUTES1, and unfolding the functions ZEROP, EQUAL, TIMES, and PLUS, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP X)) (EQUAL (TIMES (SUB1 X) (ADD1 Y)) (PLUS (SUB1 X) (TIMES (SUB1 X) Y)))) (EQUAL (TIMES X (ADD1 Y)) (PLUS X (TIMES X Y)))). This simplifies, rewriting with the lemma SUB1-ADD1, and expanding the functions ZEROP, TIMES, and PLUS, to the following two new formulas: Case 1.2. (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (TIMES (SUB1 X) (ADD1 Y)) (PLUS (SUB1 X) (TIMES (SUB1 X) Y))) (NOT (NUMBERP Y))) (EQUAL (ADD1 (PLUS 0 (TIMES (SUB1 X) (ADD1 Y)))) (PLUS X Y (TIMES (SUB1 X) Y)))). But this again simplifies, unfolding the functions EQUAL and PLUS, to: T. Case 1.1. (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (TIMES (SUB1 X) (ADD1 Y)) (PLUS (SUB1 X) (TIMES (SUB1 X) Y))) (NUMBERP Y)) (EQUAL (ADD1 (PLUS Y (TIMES (SUB1 X) (ADD1 Y)))) (PLUS X Y (TIMES (SUB1 X) Y)))), which again simplifies, using linear arithmetic, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] TIMES-ADD1 (PROVE-LEMMA TIMES-DISTRIBUTES1 (REWRITE) (EQUAL (TIMES X (PLUS Y Z)) (PLUS (TIMES X Y) (TIMES X Z)))) Call the conjecture *1. We will try to prove it by induction. There are four plausible inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (ZEROP X) (p X Y Z)) (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y Z)) (p X Y Z))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform us that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates two new goals: Case 2. (IMPLIES (ZEROP X) (EQUAL (TIMES X (PLUS Y Z)) (PLUS (TIMES X Y) (TIMES X Z)))), which simplifies, opening up the functions ZEROP, EQUAL, TIMES, and PLUS, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP X)) (EQUAL (TIMES (SUB1 X) (PLUS Y Z)) (PLUS (TIMES (SUB1 X) Y) (TIMES (SUB1 X) Z)))) (EQUAL (TIMES X (PLUS Y Z)) (PLUS (TIMES X Y) (TIMES X Z)))), which simplifies, applying PLUS-ASSOCIATES and PLUS-COMMUTES2, and unfolding the functions ZEROP and TIMES, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] TIMES-DISTRIBUTES1 (PROVE-LEMMA TIMES-COMMUTES1 (REWRITE) (EQUAL (TIMES X Y) (TIMES Y X))) WARNING: the newly proposed lemma, TIMES-COMMUTES1, could be applied whenever the previously added lemma TIMES-DISTRIBUTES1 could. WARNING: the newly proposed lemma, TIMES-COMMUTES1, could be applied whenever the previously added lemma TIMES-ADD1 could. WARNING: the newly proposed lemma, TIMES-COMMUTES1, could be applied whenever the previously added lemma TIMES-NON-NUMBERP could. WARNING: the newly proposed lemma, TIMES-COMMUTES1, could be applied whenever the previously added lemma TIMES-0 could. Give the conjecture the name *1. We will appeal to induction. Two inductions are suggested by terms in the conjecture, both of which are flawed. We limit our consideration to the two suggested by the largest number of nonprimitive recursive functions in the conjecture. Since both of these are equally likely, we will choose arbitrarily. We will induct according to the following scheme: (AND (IMPLIES (ZEROP X) (p X Y)) (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y)) (p X Y))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform us that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces the following two new conjectures: Case 2. (IMPLIES (ZEROP X) (EQUAL (TIMES X Y) (TIMES Y X))). This simplifies, applying TIMES-0 and TIMES-NON-NUMBERP, and opening up the functions ZEROP, EQUAL, and TIMES, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP X)) (EQUAL (TIMES (SUB1 X) Y) (TIMES Y (SUB1 X)))) (EQUAL (TIMES X Y) (TIMES Y X))), which simplifies, unfolding the functions ZEROP and TIMES, to: (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X) (EQUAL (TIMES (SUB1 X) Y) (TIMES Y (SUB1 X)))) (EQUAL (PLUS Y (TIMES Y (SUB1 X))) (TIMES Y X))). Appealing to the lemma SUB1-ELIM, we now replace X by (ADD1 Z) to eliminate (SUB1 X). We employ the type restriction lemma noted when SUB1 was introduced to constrain the new variable. We must thus prove: (IMPLIES (AND (NUMBERP Z) (NOT (EQUAL (ADD1 Z) 0)) (EQUAL (TIMES Z Y) (TIMES Y Z))) (EQUAL (PLUS Y (TIMES Y Z)) (TIMES Y (ADD1 Z)))). However this further simplifies, applying the lemma TIMES-ADD1, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] TIMES-COMMUTES1 (PROVE-LEMMA TIMES-COMMUTES2 (REWRITE) (EQUAL (TIMES X Y Z) (TIMES Y X Z))) WARNING: the previously added lemma, TIMES-COMMUTES1, could be applied whenever the newly proposed TIMES-COMMUTES2 could! Call the conjecture *1. Perhaps we can prove it by induction. Four inductions are suggested by terms in the conjecture. They merge into two likely candidate inductions, both of which are unflawed. Since both of these are equally likely, we will choose arbitrarily. We will induct according to the following scheme: (AND (IMPLIES (ZEROP Y) (p X Y Z)) (IMPLIES (AND (NOT (ZEROP Y)) (p X (SUB1 Y) Z)) (p X Y Z))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to prove that the measure (COUNT Y) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to two new goals: Case 2. (IMPLIES (ZEROP Y) (EQUAL (TIMES X Y Z) (TIMES Y X Z))), which simplifies, applying the lemma TIMES-COMMUTES1, and opening up the definitions of ZEROP, EQUAL, and TIMES, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP Y)) (EQUAL (TIMES X (SUB1 Y) Z) (TIMES (SUB1 Y) X Z))) (EQUAL (TIMES X Y Z) (TIMES Y X Z))), which simplifies, rewriting with TIMES-DISTRIBUTES1, and opening up the functions ZEROP and TIMES, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] TIMES-COMMUTES2 (PROVE-LEMMA TIMES-ASSOCIATES (REWRITE) (EQUAL (TIMES (TIMES X Y) Z) (TIMES X Y Z))) WARNING: the previously added lemma, TIMES-COMMUTES1, could be applied whenever the newly proposed TIMES-ASSOCIATES could! This simplifies, rewriting with TIMES-COMMUTES1 and TIMES-COMMUTES2, to: T. Q.E.D. [ 0.0 0.0 0.0 ] TIMES-ASSOCIATES (PROVE-LEMMA TIMES-DISTRIBUTES2 (REWRITE) (EQUAL (TIMES (PLUS X Y) Z) (PLUS (TIMES X Z) (TIMES Y Z)))) WARNING: the previously added lemma, TIMES-COMMUTES1, could be applied whenever the newly proposed TIMES-DISTRIBUTES2 could! This simplifies, appealing to the lemmas TIMES-COMMUTES1 and TIMES-DISTRIBUTES1, to: T. Q.E.D. [ 0.0 0.0 0.0 ] TIMES-DISTRIBUTES2 (PROVE-LEMMA DIFFERENCE-IS-0 (REWRITE) (IMPLIES (NOT (LESSP Y X)) (EQUAL (DIFFERENCE X Y) 0))) Name the conjecture *1. Let us appeal to the induction principle. The recursive terms in the conjecture suggest four inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL X 0) (NOT (NUMBERP X))) (p X Y)) (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (p X Y)) (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (p (SUB1 X) (SUB1 Y))) (p X Y))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT can be used to show that the measure (COUNT Y) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for X. The above induction scheme leads to the following four new goals: Case 4. (IMPLIES (AND (OR (EQUAL X 0) (NOT (NUMBERP X))) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)). This simplifies, expanding the definitions of NOT, OR, EQUAL, LESSP, and DIFFERENCE, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (OR (EQUAL Y 0) (NOT (NUMBERP Y))) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)). This simplifies, unfolding NOT, OR, EQUAL, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (LESSP (SUB1 Y) (SUB1 X)) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)). This simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP X 1) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (LESSP (SUB1 Y) (SUB1 X)) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)), which again simplifies, opening up the definitions of SUB1, NUMBERP, EQUAL, LESSP, NOT, and OR, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y)) 0) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)), which simplifies, using linear arithmetic, to three new conjectures: Case 1.3. (IMPLIES (AND (LESSP X Y) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y)) 0) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)), which again simplifies, using linear arithmetic, to two new conjectures: Case 1.3.2. (IMPLIES (AND (LESSP (SUB1 X) (SUB1 Y)) (LESSP X Y) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y)) 0) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)), which again simplifies, unfolding the functions LESSP, NOT, OR, DIFFERENCE, and EQUAL, to: T. Case 1.3.1. (IMPLIES (AND (LESSP X 1) (LESSP X Y) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y)) 0) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)), which again simplifies, unfolding SUB1, NUMBERP, EQUAL, LESSP, NOT, and OR, to: T. Case 1.2. (IMPLIES (AND (LESSP (SUB1 X) (SUB1 Y)) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y)) 0) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)), which again simplifies, expanding the definitions of NOT, OR, LESSP, DIFFERENCE, and EQUAL, to: T. Case 1.1. (IMPLIES (AND (LESSP X 1) (NOT (OR (EQUAL X 0) (NOT (NUMBERP X)))) (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y)))) (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y)) 0) (NOT (LESSP Y X))) (EQUAL (DIFFERENCE X Y) 0)), which again simplifies, opening up the definitions of SUB1, NUMBERP, EQUAL, LESSP, NOT, and OR, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] DIFFERENCE-IS-0 (PROVE-LEMMA DIFFERENCE-PLUS-CANCELLATION1 (REWRITE) (EQUAL (DIFFERENCE (PLUS I X) I) (FIX X))) This conjecture simplifies, opening up the function FIX, to the following two new goals: Case 2. (IMPLIES (NOT (NUMBERP X)) (EQUAL (DIFFERENCE (PLUS I X) I) 0)). But this again simplifies, using linear arithmetic, applying DIFFERENCE-IS-0, and expanding EQUAL, to: T. Case 1. (IMPLIES (NUMBERP X) (EQUAL (DIFFERENCE (PLUS I X) I) X)). However this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (PLUS I X) I) (NUMBERP X)) (EQUAL (DIFFERENCE (PLUS I X) I) X)). This again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] DIFFERENCE-PLUS-CANCELLATION1 (PROVE-LEMMA DIFFERENCE-PLUS-CANCELLATION2 (REWRITE) (EQUAL (DIFFERENCE (PLUS I X) (PLUS I Y)) (DIFFERENCE X Y))) This simplifies, using linear arithmetic, to the following two new conjectures: Case 2. (IMPLIES (LESSP (PLUS I X) (PLUS I Y)) (EQUAL (DIFFERENCE (PLUS I X) (PLUS I Y)) (DIFFERENCE X Y))). But this again simplifies, using linear arithmetic, applying DIFFERENCE-IS-0, and opening up the function EQUAL, to: T. Case 1. (IMPLIES (LESSP X Y) (EQUAL (DIFFERENCE (PLUS I X) (PLUS I Y)) (DIFFERENCE X Y))). But this again simplifies, using linear arithmetic, applying the lemma DIFFERENCE-IS-0, and expanding EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] DIFFERENCE-PLUS-CANCELLATION2 (PROVE-LEMMA DIFFERENCE-PLUS-CANCELLATION3 (REWRITE) (EQUAL (DIFFERENCE (PLUS I J X) J) (PLUS I X))) This simplifies, using linear arithmetic, to: (IMPLIES (LESSP (PLUS I J X) J) (EQUAL (DIFFERENCE (PLUS I J X) J) (PLUS I X))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] DIFFERENCE-PLUS-CANCELLATION3 (PROVE-LEMMA LESSP-REMAINDER (GENERALIZE) (EQUAL (LESSP (REMAINDER X Y) Y) (NOT (ZEROP Y)))) This conjecture simplifies, expanding ZEROP and NOT, to the following three new goals: Case 3. (IMPLIES (NOT (NUMBERP Y)) (EQUAL (LESSP (REMAINDER X Y) Y) F)). However this again simplifies, expanding the definitions of REMAINDER, LESSP, and EQUAL, to: T. Case 2. (IMPLIES (EQUAL Y 0) (EQUAL (LESSP (REMAINDER X Y) Y) F)), which again simplifies, expanding the definitions of EQUAL, REMAINDER, and LESSP, to: T. Case 1. (IMPLIES (AND (NOT (EQUAL Y 0)) (NUMBERP Y)) (EQUAL (LESSP (REMAINDER X Y) Y) T)), which again simplifies, trivially, to the new formula: (IMPLIES (AND (NOT (EQUAL Y 0)) (NUMBERP Y)) (LESSP (REMAINDER X Y) Y)), which we will name *1. We will appeal to induction. There are two plausible inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (ZEROP Y) (p X Y)) (IMPLIES (AND (NOT (ZEROP Y)) (LESSP X Y)) (p X Y)) (IMPLIES (AND (NOT (ZEROP Y)) (NOT (LESSP X Y)) (p (DIFFERENCE X Y) Y)) (p X Y))). Linear arithmetic, the lemmas COUNT-NUMBERP and COUNT-NOT-LESSP, and the definition of ZEROP establish that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces the following three new formulas: Case 3. (IMPLIES (AND (ZEROP Y) (NOT (EQUAL Y 0)) (NUMBERP Y)) (LESSP (REMAINDER X Y) Y)). This simplifies, opening up ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (ZEROP Y)) (LESSP X Y) (NOT (EQUAL Y 0)) (NUMBERP Y)) (LESSP (REMAINDER X Y) Y)). This simplifies, opening up the definitions of ZEROP and REMAINDER, to: (IMPLIES (AND (LESSP X Y) (NOT (EQUAL Y 0)) (NUMBERP Y) (NOT (NUMBERP X))) (LESSP 0 Y)), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP Y)) (NOT (LESSP X Y)) (LESSP (REMAINDER (DIFFERENCE X Y) Y) Y) (NOT (EQUAL Y 0)) (NUMBERP Y)) (LESSP (REMAINDER X Y) Y)), which simplifies, unfolding the functions ZEROP and REMAINDER, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LESSP-REMAINDER (PROVE-LEMMA REMAINDER-QUOTIENT-ELIM (ELIM REWRITE) (IMPLIES (NUMBERP X) (EQUAL (PLUS (REMAINDER X Y) (TIMES Y (QUOTIENT X Y))) X))) WARNING: the previously added lemma, PLUS-COMMUTES1, could be applied whenever the newly proposed REMAINDER-QUOTIENT-ELIM could! Give the conjecture the name *1. Let us appeal to the induction principle. The recursive terms in the conjecture suggest three inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (ZEROP Y) (p X Y)) (IMPLIES (AND (NOT (ZEROP Y)) (LESSP X Y)) (p X Y)) (IMPLIES (AND (NOT (ZEROP Y)) (NOT (LESSP X Y)) (p (DIFFERENCE X Y) Y)) (p X Y))). Linear arithmetic, the lemmas COUNT-NUMBERP and COUNT-NOT-LESSP, and the definition of ZEROP can be used to show that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates the following three new conjectures: Case 3. (IMPLIES (AND (ZEROP Y) (NUMBERP X)) (EQUAL (PLUS (REMAINDER X Y) (TIMES Y (QUOTIENT X Y))) X)). This simplifies, appealing to the lemmas PLUS-COMMUTES1, TIMES-NON-NUMBERP, and TIMES-COMMUTES1, and expanding the functions ZEROP, EQUAL, REMAINDER, QUOTIENT, TIMES, and PLUS, to: T. Case 2. (IMPLIES (AND (NOT (ZEROP Y)) (LESSP X Y) (NUMBERP X)) (EQUAL (PLUS (REMAINDER X Y) (TIMES Y (QUOTIENT X Y))) X)). This simplifies, applying TIMES-COMMUTES1 and PLUS-COMMUTES1, and expanding the definitions of ZEROP, REMAINDER, QUOTIENT, EQUAL, TIMES, and PLUS, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP Y)) (NOT (LESSP X Y)) (EQUAL (PLUS (REMAINDER (DIFFERENCE X Y) Y) (TIMES Y (QUOTIENT (DIFFERENCE X Y) Y))) (DIFFERENCE X Y)) (NUMBERP X)) (EQUAL (PLUS (REMAINDER X Y) (TIMES Y (QUOTIENT X Y))) X)), which simplifies, applying TIMES-ADD1 and PLUS-COMMUTES2, and opening up ZEROP, REMAINDER, and QUOTIENT, to: (IMPLIES (AND (NOT (EQUAL Y 0)) (NUMBERP Y) (NOT (LESSP X Y)) (EQUAL (PLUS (REMAINDER (DIFFERENCE X Y) Y) (TIMES Y (QUOTIENT (DIFFERENCE X Y) Y))) (DIFFERENCE X Y)) (NUMBERP X)) (EQUAL (PLUS Y (DIFFERENCE X Y)) X)), which again simplifies, using linear arithmetic, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] REMAINDER-QUOTIENT-ELIM (DEFN TIC-TAC-TOE1-A (XMOVES BOARD I) (LET ((BOARD1 (MOVE 'X (CAR XMOVES) BOARD))) (IF (LEGALP (CAR XMOVES) BOARD) (IF (LESSP I 9) (LET ((BOARD2 (MOVE 'O (PICK-MOVE BOARD1) BOARD1))) (IF (WINP 'O BOARD2) (LIST BOARD1 BOARD2 'O-WIN) (CONS BOARD1 (CONS BOARD2 (TIC-TAC-TOE1-A (CDR XMOVES) BOARD2 (ADD1 (ADD1 I))))))) (LIST BOARD1 'DRAW)) (LIST BOARD1 'O-WIN))) ((LESSP (DIFFERENCE 9 I)))) Linear arithmetic, the lemma SUB1-TYPE-RESTRICTION, and the definitions of LESSP, ADD1, and DIFFERENCE establish that the measure (DIFFERENCE 9 I) decreases according to the well-founded relation LESSP in each recursive call. Hence, TIC-TAC-TOE1-A is accepted under the principle of definition. Note that (LISTP (TIC-TAC-TOE1-A XMOVES BOARD I)) is a theorem. [ 0.1 0.0 0.0 ] TIC-TAC-TOE1-A (DEFN TIC-TAC-TOE-A (XMOVES) (LET ((BOARD0 (LIST F F F F F F F F F))) (CONS BOARD0 (TIC-TAC-TOE1-A XMOVES BOARD0 1)))) Note that (LISTP (TIC-TAC-TOE-A XMOVES)) is a theorem. [ 0.0 0.0 0.0 ] TIC-TAC-TOE-A (ENABLE TIC-TAC-TOE1) [ 0.0 0.0 0.0 ] TIC-TAC-TOE1-ON (ENABLE LEGALP) [ 0.0 0.0 0.0 ] LEGALP-ON (ENABLE MOVE) [ 0.0 0.0 0.0 ] MOVE-ON (PROVE-LEMMA ENDP-OPEN-SQUARES (REWRITE) (EQUAL (ENDP BOARD) (ZEROP (OPEN-SQUARES BOARD)))) This formula simplifies, unfolding the definition of ZEROP, to the following two new goals: Case 2. (IMPLIES (NOT (EQUAL (OPEN-SQUARES BOARD) 0)) (NOT (ENDP BOARD))). Name the above subgoal *1. Case 1. (IMPLIES (EQUAL (OPEN-SQUARES BOARD) 0) (EQUAL (ENDP BOARD) T)). This again simplifies, trivially, to the new goal: (IMPLIES (EQUAL (OPEN-SQUARES BOARD) 0) (ENDP BOARD)), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (EQUAL (ENDP BOARD) (ZEROP (OPEN-SQUARES BOARD))), named *1. Let us appeal to the induction principle. The recursive terms in the conjecture suggest two inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (p (CDR BOARD))) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (p (CDR BOARD))) (p BOARD))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to three new goals: Case 3. (IMPLIES (NLISTP BOARD) (EQUAL (ENDP BOARD) (ZEROP (OPEN-SQUARES BOARD)))), which simplifies, opening up the definitions of NLISTP, ENDP, OPEN-SQUARES, ZEROP, and EQUAL, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (EQUAL (ENDP (CDR BOARD)) (ZEROP (OPEN-SQUARES (CDR BOARD))))) (EQUAL (ENDP BOARD) (ZEROP (OPEN-SQUARES BOARD)))), which simplifies, unfolding the functions NLISTP, ZEROP, ENDP, OPEN-SQUARES, and EQUAL, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (EQUAL (ENDP (CDR BOARD)) (ZEROP (OPEN-SQUARES (CDR BOARD))))) (EQUAL (ENDP BOARD) (ZEROP (OPEN-SQUARES BOARD)))), which simplifies, expanding the definitions of NLISTP, ZEROP, ENDP, OPEN-SQUARES, EQUAL, and ADD1, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] ENDP-OPEN-SQUARES (PROVE-LEMMA OPEN-SQUARES-MOVE-REWRITE (REWRITE) (IMPLIES (AND X (LEGALP I BOARD)) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD))))) This conjecture can be simplified, using the abbreviations LEGALP, AND, and IMPLIES, to: (IMPLIES (AND X (LESSP I 9) (EQUAL (NTH I BOARD) F)) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD)))). This simplifies, trivially, to: (IMPLIES (AND X (LESSP I 9) (NOT (NTH I BOARD))) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD)))), which we will name *1. Perhaps we can prove it by induction. There are five plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL 9 0) (NOT (NUMBERP 9))) (p X I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p X I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p X (SUB1 I) (CDR BOARD))) (p X I BOARD))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT can be used to establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme produces five new formulas: Case 5. (IMPLIES (AND (OR (EQUAL 9 0) (NOT (NUMBERP 9))) X (LESSP I 9) (NOT (NTH I BOARD))) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD)))), which simplifies, expanding the definitions of EQUAL, NUMBERP, NOT, and OR, to: T. Case 4. (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (OR (EQUAL I 0) (NOT (NUMBERP I))) X (LESSP I 9) (NOT (NTH I BOARD))) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD)))), which simplifies, expanding EQUAL, NUMBERP, NOT, OR, LESSP, NTH, MOVE, and OPEN-SQUARES, to four new conjectures: Case 4.4. (IMPLIES (AND (EQUAL I 0) X (NOT (CAR BOARD)) (NOT (LISTP BOARD))) (EQUAL (OPEN-SQUARES BOARD) (SUB1 0))), which again simplifies, rewriting with CAR-NLISTP, to: T. Case 4.3. (IMPLIES (AND (EQUAL I 0) X (NOT (CAR BOARD)) (LISTP BOARD)) (EQUAL (OPEN-SQUARES (CONS X (CDR BOARD))) (SUB1 (ADD1 (OPEN-SQUARES (CDR BOARD)))))). But this again simplifies, applying CDR-CONS, CAR-CONS, and SUB1-ADD1, and expanding the definition of OPEN-SQUARES, to: T. Case 4.2. (IMPLIES (AND (NOT (NUMBERP I)) X (NOT (CAR BOARD)) (NOT (LISTP BOARD))) (EQUAL (OPEN-SQUARES BOARD) (SUB1 0))). This again simplifies, applying the lemma CAR-NLISTP, to: T. Case 4.1. (IMPLIES (AND (NOT (NUMBERP I)) X (NOT (CAR BOARD)) (LISTP BOARD)) (EQUAL (OPEN-SQUARES (CONS X (CDR BOARD))) (SUB1 (ADD1 (OPEN-SQUARES (CDR BOARD)))))), which again simplifies, applying the lemmas CDR-CONS, CAR-CONS, and SUB1-ADD1, and unfolding OPEN-SQUARES, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (LESSP (SUB1 I) 9)) X (LESSP I 9) (NOT (NTH I BOARD))) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD)))), which simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP I 1) (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (LESSP (SUB1 I) 9)) X (LESSP I 9) (NOT (NTH I BOARD))) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD)))). However this again simplifies, unfolding the functions SUB1, NUMBERP, EQUAL, LESSP, NOT, and OR, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NTH (SUB1 I) (CDR BOARD)) X (LESSP I 9) (NOT (NTH I BOARD))) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD)))), which simplifies, opening up the definitions of EQUAL, NUMBERP, NOT, OR, LESSP, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL 9 0) (NOT (NUMBERP 9)))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (EQUAL (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (SUB1 (OPEN-SQUARES (CDR BOARD)))) X (LESSP I 9) (NOT (NTH I BOARD))) (EQUAL (OPEN-SQUARES (MOVE X I BOARD)) (SUB1 (OPEN-SQUARES BOARD)))), which simplifies, expanding the definitions of EQUAL, NUMBERP, NOT, OR, LESSP, SUB1, NTH, MOVE, and OPEN-SQUARES, to three new conjectures: Case 1.3. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (SUB1 (OPEN-SQUARES (CDR BOARD)))) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD))) (NOT (LISTP BOARD))) (EQUAL (OPEN-SQUARES BOARD) (SUB1 0))), which again simplifies, opening up the definitions of OPEN-SQUARES, SUB1, and EQUAL, to: T. Case 1.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (SUB1 (OPEN-SQUARES (CDR BOARD)))) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD))) (LISTP BOARD) (CAR BOARD)) (EQUAL (OPEN-SQUARES (CONS (CAR BOARD) (MOVE X (SUB1 I) (CDR BOARD)))) (SUB1 (OPEN-SQUARES (CDR BOARD))))), which again simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding OPEN-SQUARES, to: T. Case 1.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (SUB1 (OPEN-SQUARES (CDR BOARD)))) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD))) (LISTP BOARD) (NOT (CAR BOARD))) (EQUAL (OPEN-SQUARES (CONS (CAR BOARD) (MOVE X (SUB1 I) (CDR BOARD)))) (SUB1 (ADD1 (OPEN-SQUARES (CDR BOARD)))))). However this again simplifies, rewriting with the lemmas ADD1-SUB1, CDR-CONS, CAR-CONS, and SUB1-ADD1, and unfolding the definition of OPEN-SQUARES, to the formula: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) (SUB1 (OPEN-SQUARES (CDR BOARD)))) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD))) (LISTP BOARD) (NOT (CAR BOARD)) (EQUAL (OPEN-SQUARES (CDR BOARD)) 0)) (EQUAL 1 (OPEN-SQUARES (CDR BOARD)))). This again simplifies, opening up the functions SUB1 and EQUAL, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (OPEN-SQUARES (MOVE X (SUB1 I) (CDR BOARD))) 0) X (LESSP (SUB1 I) 8) (NOT (NTH (SUB1 I) (CDR BOARD))) (LISTP BOARD) (NOT (CAR BOARD))) (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))). Appealing to the lemma SUB1-ELIM, we now replace I by (ADD1 Z) to eliminate (SUB1 I). We employ the type restriction lemma noted when SUB1 was introduced to constrain the new variable. We must thus prove the goal: (IMPLIES (AND (NUMBERP Z) (NOT (EQUAL (ADD1 Z) 0)) (EQUAL (OPEN-SQUARES (MOVE X Z (CDR BOARD))) 0) X (LESSP Z 8) (NOT (NTH Z (CDR BOARD))) (LISTP BOARD) (NOT (CAR BOARD))) (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))). This further simplifies, clearly, to the new formula: (IMPLIES (AND (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z (CDR BOARD))) 0) X (LESSP Z 8) (NOT (NTH Z (CDR BOARD))) (LISTP BOARD) (NOT (CAR BOARD))) (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))). Applying the lemma CAR-CDR-ELIM, replace BOARD by (CONS W V) to eliminate (CDR BOARD) and (CAR BOARD). This produces: (IMPLIES (AND (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V)) (NOT W)) (NOT (EQUAL (OPEN-SQUARES V) 0))), which further simplifies, obviously, to: (IMPLIES (AND (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V))) (NOT (EQUAL (OPEN-SQUARES V) 0))), which we will finally name *1.1. Perhaps we can prove it by induction. There are five plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP V) (p V Z X)) (IMPLIES (AND (NOT (NLISTP V)) (ZEROP Z)) (p V Z X)) (IMPLIES (AND (NOT (NLISTP V)) (NOT (ZEROP Z)) (p (CDR V) (SUB1 Z) X)) (p V Z X))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP can be used to establish that the measure (COUNT Z) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for V. The above induction scheme produces six new formulas: Case 6. (IMPLIES (AND (NLISTP V) (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V))) (NOT (EQUAL (OPEN-SQUARES V) 0))), which simplifies, appealing to the lemma CAR-NLISTP, and unfolding the functions NLISTP, MOVE, OPEN-SQUARES, EQUAL, LESSP, NUMBERP, SUB1, and NTH, to: (IMPLIES (AND (NOT (LISTP V)) (NUMBERP Z) X (LESSP (SUB1 Z) 7) (NOT (EQUAL Z 0))) (NTH (SUB1 Z) (CDR V))). But this further simplifies, applying CDR-NLISTP, to: (IMPLIES (AND (NOT (LISTP V)) (NUMBERP Z) X (LESSP (SUB1 Z) 7) (NOT (EQUAL Z 0))) (NTH (SUB1 Z) 0)). Applying the lemma SUB1-ELIM, replace Z by (ADD1 W) to eliminate (SUB1 Z). We rely upon the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We thus obtain: (IMPLIES (AND (NUMBERP W) (NOT (LISTP V)) X (LESSP W 7) (NOT (EQUAL (ADD1 W) 0))) (NTH W 0)), which further simplifies, clearly, to: (IMPLIES (AND (NUMBERP W) (NOT (LISTP V)) X (LESSP W 7)) (NTH W 0)), which has two irrelevant terms in it. By eliminating these terms we get the new goal: (IMPLIES (AND (NUMBERP W) (LESSP W 7)) (NTH W 0)), which we will finally name *1.1.1. Case 5. (IMPLIES (AND (NOT (NLISTP V)) (ZEROP Z) (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V))) (NOT (EQUAL (OPEN-SQUARES V) 0))). This simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and expanding the functions NLISTP, ZEROP, NUMBERP, EQUAL, MOVE, OPEN-SQUARES, LESSP, NTH, and ADD1, to: T. Case 4. (IMPLIES (AND (NOT (NLISTP V)) (NOT (ZEROP Z)) (NOT (EQUAL (OPEN-SQUARES (MOVE X (SUB1 Z) (CDR V))) 0)) (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V))) (NOT (EQUAL (OPEN-SQUARES V) 0))). This simplifies, applying CDR-CONS and CAR-CONS, and expanding the definitions of NLISTP, ZEROP, MOVE, OPEN-SQUARES, LESSP, EQUAL, NUMBERP, SUB1, and NTH, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP V)) (NOT (ZEROP Z)) (NOT (LESSP (SUB1 Z) 8)) (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V))) (NOT (EQUAL (OPEN-SQUARES V) 0))), which simplifies, using linear arithmetic, to: (IMPLIES (AND (EQUAL Z 0) (NOT (NLISTP V)) (NOT (ZEROP Z)) (NOT (LESSP (SUB1 Z) 8)) (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V))) (NOT (EQUAL (OPEN-SQUARES V) 0))). But this again simplifies, expanding the functions NLISTP and ZEROP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP V)) (NOT (ZEROP Z)) (NTH (SUB1 Z) (CDR V)) (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V))) (NOT (EQUAL (OPEN-SQUARES V) 0))), which simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding the definitions of NLISTP, ZEROP, MOVE, OPEN-SQUARES, LESSP, EQUAL, NUMBERP, SUB1, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP V)) (NOT (ZEROP Z)) (NOT (EQUAL (OPEN-SQUARES (CDR V)) 0)) (NUMBERP Z) (EQUAL (OPEN-SQUARES (MOVE X Z V)) 0) X (LESSP Z 8) (NOT (NTH Z V))) (NOT (EQUAL (OPEN-SQUARES V) 0))), which simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up the definitions of NLISTP, ZEROP, MOVE, OPEN-SQUARES, LESSP, EQUAL, NUMBERP, SUB1, and NTH, to: T. So let us turn our attention to: (IMPLIES (AND (NUMBERP W) (LESSP W 7)) (NTH W 0)), which is formula *1.1.1 above. Let us appeal to the induction principle. Two inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL 7 0) (NOT (NUMBERP 7))) (p W)) (IMPLIES (AND (NOT (OR (EQUAL 7 0) (NOT (NUMBERP 7)))) (OR (EQUAL W 0) (NOT (NUMBERP W)))) (p W)) (IMPLIES (AND (NOT (OR (EQUAL 7 0) (NOT (NUMBERP 7)))) (NOT (OR (EQUAL W 0) (NOT (NUMBERP W)))) (p (SUB1 W))) (p W))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT can be used to establish that the measure (COUNT W) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to the following four new goals: Case 4. (IMPLIES (AND (OR (EQUAL 7 0) (NOT (NUMBERP 7))) (NUMBERP W) (LESSP W 7)) (NTH W 0)). This simplifies, unfolding EQUAL, NUMBERP, NOT, and OR, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL 7 0) (NOT (NUMBERP 7)))) (OR (EQUAL W 0) (NOT (NUMBERP W))) (NUMBERP W) (LESSP W 7)) (NTH W 0)). This simplifies, opening up EQUAL, NUMBERP, NOT, OR, LESSP, and NTH, to: (IMPLIES (EQUAL W 0) 0), which again simplifies, obviously, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL 7 0) (NOT (NUMBERP 7)))) (NOT (OR (EQUAL W 0) (NOT (NUMBERP W)))) (NOT (LESSP (SUB1 W) 7)) (NUMBERP W) (LESSP W 7)) (NTH W 0)). This simplifies, using linear arithmetic, to: (IMPLIES (AND (EQUAL W 0) (NOT (OR (EQUAL 7 0) (NOT (NUMBERP 7)))) (NOT (OR (EQUAL W 0) (NOT (NUMBERP W)))) (NOT (LESSP (SUB1 W) 7)) (NUMBERP W) (LESSP W 7)) (NTH W 0)), which again simplifies, opening up EQUAL, NUMBERP, NOT, and OR, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL 7 0) (NOT (NUMBERP 7)))) (NOT (OR (EQUAL W 0) (NOT (NUMBERP W)))) (NTH (SUB1 W) 0) (NUMBERP W) (LESSP W 7)) (NTH W 0)), which simplifies, expanding EQUAL, NUMBERP, NOT, OR, LESSP, SUB1, NTH, and CDR, to: T. That finishes the proof of *1.1.1, which, consequently, finishes the proof of *1.1, which, in turn, finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.0 ] OPEN-SQUARES-MOVE-REWRITE (PROVE-LEMMA ADD1-DIFFERENCE (REWRITE) (IMPLIES (NOT (LESSP X Y)) (EQUAL (ADD1 (DIFFERENCE X Y)) (DIFFERENCE (ADD1 X) Y)))) This simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (ADD1 X) Y) (NOT (LESSP X Y))) (EQUAL (ADD1 (DIFFERENCE X Y)) (DIFFERENCE (ADD1 X) Y))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] ADD1-DIFFERENCE (PROVE-LEMMA LESSP-OPEN-SQUARES-LENGTH (REWRITE) (NOT (LESSP (LENGTH BOARD) (OPEN-SQUARES BOARD)))) WARNING: Note that the proposed lemma LESSP-OPEN-SQUARES-LENGTH is to be stored as zero type prescription rules, zero compound recognizer rules, two linear rules, and zero replacement rules. Give the conjecture the name *1. We will appeal to induction. Two inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (p (CDR BOARD))) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (p (CDR BOARD))) (p BOARD))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces the following three new conjectures: Case 3. (IMPLIES (NLISTP BOARD) (NOT (LESSP (LENGTH BOARD) (OPEN-SQUARES BOARD)))). This simplifies, expanding the functions NLISTP, LENGTH, OPEN-SQUARES, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (CAR BOARD) (NOT (LESSP (LENGTH (CDR BOARD)) (OPEN-SQUARES (CDR BOARD))))) (NOT (LESSP (LENGTH BOARD) (OPEN-SQUARES BOARD)))). This simplifies, rewriting with the lemma SUB1-ADD1, and opening up NLISTP, LENGTH, OPEN-SQUARES, and LESSP, to: (IMPLIES (AND (LISTP BOARD) (CAR BOARD) (NOT (LESSP (LENGTH (CDR BOARD)) (OPEN-SQUARES (CDR BOARD)))) (NOT (EQUAL (OPEN-SQUARES (CDR BOARD)) 0))) (NOT (LESSP (LENGTH (CDR BOARD)) (SUB1 (OPEN-SQUARES (CDR BOARD)))))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (CAR BOARD)) (NOT (LESSP (LENGTH (CDR BOARD)) (OPEN-SQUARES (CDR BOARD))))) (NOT (LESSP (LENGTH BOARD) (OPEN-SQUARES BOARD)))), which simplifies, applying SUB1-ADD1, and opening up NLISTP, LENGTH, OPEN-SQUARES, and LESSP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.1 ] LESSP-OPEN-SQUARES-LENGTH (PROVE-LEMMA LEMMA-A1 NIL (IMPLIES (EQUAL (LENGTH BOARD) 9) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (TIC-TAC-TOE1-A XMOVES BOARD (DIFFERENCE 10 (OPEN-SQUARES BOARD))))) ((EXPAND (TIC-TAC-TOE1-A XMOVES BOARD (DIFFERENCE 10 (OPEN-SQUARES BOARD)))))) This conjecture simplifies, using linear arithmetic, applying LESSP-OPEN-SQUARES-LENGTH and ADD1-DIFFERENCE, and opening up the definitions of TIC-TAC-TOE1-A, LEGALP, and ADD1, to five new formulas: Case 5. (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))))) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (CONS (MOVE 'X (CAR XMOVES) BOARD) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 12 (OPEN-SQUARES BOARD))))))). Applying the lemma CAR-CDR-ELIM, replace XMOVES by (CONS X Z) to eliminate (CAR XMOVES) and (CDR XMOVES). We would thus like to prove the following two new conjectures: Case 5.2. (IMPLIES (AND (NOT (LISTP XMOVES)) (EQUAL (LENGTH BOARD) 9) (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))))) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (CONS (MOVE 'X (CAR XMOVES) BOARD) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 12 (OPEN-SQUARES BOARD))))))). But this further simplifies, rewriting with CAR-NLISTP, CDR-NLISTP, CAR-CONS, CDR-CONS, and CONS-EQUAL, and unfolding LESSP, EQUAL, NTH, MOVE, LEGALP, TIC-TAC-TOE1, OPEN-SQUARES, DIFFERENCE, CAR, and TIC-TAC-TOE1-A, to the following four new formulas: Case 5.2.4. (IMPLIES (AND (NOT (LISTP XMOVES)) (EQUAL (LENGTH BOARD) 9) (NOT (CAR BOARD)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9) (NOT (LISTP BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE BOARD) BOARD))). But this again simplifies, opening up the definitions of LENGTH and EQUAL, to: T. Case 5.2.3. (IMPLIES (AND (NOT (LISTP XMOVES)) (EQUAL (LENGTH BOARD) 9) (NOT (CAR BOARD)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9) (LISTP BOARD) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (CONS 'X (CDR BOARD))) (CONS 'X (CDR BOARD))))) (NOT (EQUAL (PICK-MOVE (CONS 'X (CDR BOARD))) 0)) (NUMBERP (PICK-MOVE (CONS 'X (CDR BOARD))))) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (CONS (CONS 'X (CDR BOARD)) (CONS (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD))) (TIC-TAC-TOE1-A 0 (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD))) (DIFFERENCE 12 (OPEN-SQUARES BOARD))))))), which again simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding the functions MOVE, LEGALP, LESSP, EQUAL, NTH, CAR, and TIC-TAC-TOE1-A, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (EQUAL (LENGTH BOARD) 9) (NOT (CAR BOARD)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9) (LISTP BOARD) (NOT (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD))))) (NOT (EQUAL (PICK-MOVE (CONS 'X (CDR BOARD))) 0)) (NUMBERP (PICK-MOVE (CONS 'X (CDR BOARD))))) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (CONS (CONS 'X (CDR BOARD)) (CONS (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD))) (CONS (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD))) '(O-WIN)))))). Appealing to the lemma CAR-CDR-ELIM, we now replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). This generates: (IMPLIES (AND (NOT (LISTP XMOVES)) (EQUAL (LENGTH (CONS X Z)) 9) (NOT X) (LESSP (DIFFERENCE 10 (OPEN-SQUARES (CONS X Z))) 9) (NOT (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)))) (NOT (EQUAL (PICK-MOVE (CONS 'X Z)) 0)) (NUMBERP (PICK-MOVE (CONS 'X Z)))) (EQUAL (TIC-TAC-TOE1 XMOVES (CONS X Z)) (CONS (CONS 'X Z) (CONS (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)) (CONS (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)) '(O-WIN)))))). However this further simplifies, rewriting with CDR-CONS, ADD1-EQUAL, CAR-CONS, and SUB1-ADD1, and unfolding LENGTH, NUMBERP, OPEN-SQUARES, SUB1, EQUAL, and DIFFERENCE, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (EQUAL (LENGTH Z) 8) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9) (NOT (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)))) (NOT (EQUAL (PICK-MOVE (CONS 'X Z)) 0)) (NUMBERP (PICK-MOVE (CONS 'X Z)))) (EQUAL (TIC-TAC-TOE1 XMOVES (CONS F Z)) (CONS (CONS 'X Z) (CONS (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)) (CONS (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)) '(O-WIN)))))), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (IMPLIES (EQUAL (LENGTH BOARD) 9) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (TIC-TAC-TOE1-A XMOVES BOARD (DIFFERENCE 10 (OPEN-SQUARES BOARD))))). We named this *1. We will try to prove it by induction. The recursive terms in the conjecture suggest three inductions. They merge into two likely candidate inductions, both of which are unflawed. So we will choose the one suggested by the largest number of nonprimitive recursive functions. We will induct according to the following scheme: (AND (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (p XMOVES BOARD)) (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (p XMOVES BOARD)) (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (p (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (p XMOVES BOARD)) (IMPLIES (NOT (LEGALP (CAR XMOVES) BOARD)) (p XMOVES BOARD))). Linear arithmetic and the lemmas MOVE-NEVER-INCREASES-OPEN-SQUARES and LEGAL-MOVES-REDUCE-OPEN-SQUARES establish that the measure (OPEN-SQUARES BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for XMOVES. The above induction scheme leads to five new goals: Case 5. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (ENDP (MOVE 'X (CAR XMOVES) BOARD)) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (TIC-TAC-TOE1-A XMOVES BOARD (DIFFERENCE 10 (OPEN-SQUARES BOARD))))), which simplifies, applying OPEN-SQUARES-MOVE-REWRITE, ENDP-OPEN-SQUARES, and CAR-CONS, and unfolding LEGALP, EQUAL, TIC-TAC-TOE1, DIFFERENCE, SUB1, and NUMBERP, to the following two new formulas: Case 5.2. (IMPLIES (AND (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0) (EQUAL (LENGTH BOARD) 9) (NOT (EQUAL (OPEN-SQUARES BOARD) 0))) (EQUAL (CONS (MOVE 'X (CAR XMOVES) BOARD) '(DRAW)) (TIC-TAC-TOE1-A XMOVES BOARD 9))). This again simplifies, opening up the definitions of LESSP, LEGALP, EQUAL, and TIC-TAC-TOE1-A, to: T. Case 5.1. (IMPLIES (AND (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0) (EQUAL (LENGTH BOARD) 9) (EQUAL (OPEN-SQUARES BOARD) 0)) (EQUAL (CONS (MOVE 'X (CAR XMOVES) BOARD) '(DRAW)) (TIC-TAC-TOE1-A XMOVES BOARD 10))), which again simplifies, expanding SUB1, EQUAL, LESSP, LEGALP, and TIC-TAC-TOE1-A, to: T. Case 4. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (TIC-TAC-TOE1-A XMOVES BOARD (DIFFERENCE 10 (OPEN-SQUARES BOARD))))), which simplifies, applying OPEN-SQUARES-MOVE-REWRITE, ENDP-OPEN-SQUARES, and CAR-CONS, and unfolding LEGALP, EQUAL, TIC-TAC-TOE1, and TIC-TAC-TOE1-A, to: (IMPLIES (AND (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (EQUAL (LENGTH BOARD) 9) (NOT (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9))) (EQUAL (CONS (MOVE 'X (CAR XMOVES) BOARD) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) '(O-WIN))) (CONS (MOVE 'X (CAR XMOVES) BOARD) '(DRAW)))), which again simplifies, rewriting with CAR-CONS, CDR-CONS, and CONS-EQUAL, and expanding the functions CAR and EQUAL, to: (IMPLIES (AND (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). Applying the lemma CAR-CDR-ELIM, replace XMOVES by (CONS X Z) to eliminate (CAR XMOVES) and (CDR XMOVES). We would thus like to prove the following two new conjectures: Case 4.2. (IMPLIES (AND (NOT (LISTP XMOVES)) (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). However this further simplifies, applying CAR-NLISTP, and expanding the definitions of LESSP, EQUAL, NTH, MOVE, and LENGTH, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (WINP 'O (MOVE 'O (PICK-MOVE (CONS 'X (CDR BOARD))) (CONS 'X (CDR BOARD)))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)), which again simplifies, rewriting with CAR-CONS and CDR-CONS, and unfolding the definition of MOVE, to the following three new conjectures: Case 4.2.3. (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (NOT (EQUAL (PICK-MOVE (CONS 'X (CDR BOARD))) 0)) (NUMBERP (PICK-MOVE (CONS 'X (CDR BOARD)))) (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD)))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). Appealing to the lemma CAR-CDR-ELIM, we now replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). The result is: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT X) (NOT (EQUAL (SUB1 (OPEN-SQUARES (CONS X Z))) 0)) (NOT (EQUAL (PICK-MOVE (CONS 'X Z)) 0)) (NUMBERP (PICK-MOVE (CONS 'X Z))) (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z))) (EQUAL (LENGTH (CONS X Z)) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES (CONS X Z))) 9)). But this further simplifies, applying CDR-CONS, CAR-CONS, SUB1-ADD1, and ADD1-EQUAL, and unfolding the functions OPEN-SQUARES, LENGTH, NUMBERP, SUB1, EQUAL, and DIFFERENCE, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (EQUAL (PICK-MOVE (CONS 'X Z)) 0)) (NUMBERP (PICK-MOVE (CONS 'X Z))) (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z))) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)), which again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP 9 (OPEN-SQUARES Z)) (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (EQUAL (PICK-MOVE (CONS 'X Z)) 0)) (NUMBERP (PICK-MOVE (CONS 'X Z))) (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z))) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)). This finally simplifies, using linear arithmetic and rewriting with LESSP-OPEN-SQUARES-LENGTH, to: T. Case 4.2.2. (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (EQUAL (PICK-MOVE (CONS 'X (CDR BOARD))) 0) (WINP 'O (CONS 'O (CDR BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). Appealing to the lemma CAR-CDR-ELIM, we now replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). This generates the formula: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT X) (NOT (EQUAL (SUB1 (OPEN-SQUARES (CONS X Z))) 0)) (EQUAL (PICK-MOVE (CONS 'X Z)) 0) (WINP 'O (CONS 'O Z)) (EQUAL (LENGTH (CONS X Z)) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES (CONS X Z))) 9)). But this further simplifies, applying CDR-CONS, CAR-CONS, SUB1-ADD1, and ADD1-EQUAL, and expanding the definitions of OPEN-SQUARES, LENGTH, NUMBERP, SUB1, EQUAL, and DIFFERENCE, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (EQUAL (PICK-MOVE (CONS 'X Z)) 0) (WINP 'O (CONS 'O Z)) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)), which again simplifies, using linear arithmetic, to the conjecture: (IMPLIES (AND (LESSP 9 (OPEN-SQUARES Z)) (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (EQUAL (PICK-MOVE (CONS 'X Z)) 0) (WINP 'O (CONS 'O Z)) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)). This finally simplifies, using linear arithmetic and applying LESSP-OPEN-SQUARES-LENGTH, to: T. Case 4.2.1. (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (NOT (NUMBERP (PICK-MOVE (CONS 'X (CDR BOARD))))) (WINP 'O (CONS 'O (CDR BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). Appealing to the lemma CAR-CDR-ELIM, we now replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). The result is: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT X) (NOT (EQUAL (SUB1 (OPEN-SQUARES (CONS X Z))) 0)) (NOT (NUMBERP (PICK-MOVE (CONS 'X Z)))) (WINP 'O (CONS 'O Z)) (EQUAL (LENGTH (CONS X Z)) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES (CONS X Z))) 9)). However this further simplifies, rewriting with CDR-CONS, CAR-CONS, SUB1-ADD1, and ADD1-EQUAL, and opening up the definitions of OPEN-SQUARES, LENGTH, NUMBERP, SUB1, EQUAL, and DIFFERENCE, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (NUMBERP (PICK-MOVE (CONS 'X Z)))) (WINP 'O (CONS 'O Z)) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)), which again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP 9 (OPEN-SQUARES Z)) (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (NUMBERP (PICK-MOVE (CONS 'X Z)))) (WINP 'O (CONS 'O Z)) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)). This finally simplifies, using linear arithmetic and appealing to the lemma LESSP-OPEN-SQUARES-LENGTH, to: T. Case 4.1. (IMPLIES (AND (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)), which we generalize by replacing (OPEN-SQUARES BOARD) by Y. We restrict the new variable by recalling the type restriction lemma noted when OPEN-SQUARES was introduced. We would thus like to prove: (IMPLIES (AND (NUMBERP Y) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 Y) 0)) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 Y) 9)). Applying the lemma SUB1-ELIM, replace Y by (ADD1 Z) to eliminate (SUB1 Y). We use the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove the following two new conjectures: Case 4.1.2. (IMPLIES (AND (EQUAL Y 0) (NUMBERP Y) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 Y) 0)) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 Y) 9)). This finally simplifies, unfolding NUMBERP, SUB1, and EQUAL, to: T. Case 4.1.1. (IMPLIES (AND (NUMBERP Z) (NOT (EQUAL (ADD1 Z) 0)) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL Z 0)) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (ADD1 Z)) 9)), which further simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP 10 (ADD1 Z)) (NUMBERP Z) (NOT (EQUAL (ADD1 Z) 0)) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL Z 0)) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (ADD1 Z)) 9)). But this finally simplifies, using linear arithmetic, appealing to the lemmas SUB1-ADD1 and DIFFERENCE-IS-0, and opening up SUB1, NUMBERP, EQUAL, and LESSP, to: T. Case 3. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (NOT (EQUAL (LENGTH (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) 9)) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (TIC-TAC-TOE1-A XMOVES BOARD (DIFFERENCE 10 (OPEN-SQUARES BOARD))))), which simplifies, rewriting with OPEN-SQUARES-MOVE-REWRITE, ENDP-OPEN-SQUARES, and LENGTH-MOVE, and expanding LEGALP and EQUAL, to: T. Case 2. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (ENDP (MOVE 'X (CAR XMOVES) BOARD))) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 10 (OPEN-SQUARES (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))))) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (TIC-TAC-TOE1-A XMOVES BOARD (DIFFERENCE 10 (OPEN-SQUARES BOARD))))). This simplifies, using linear arithmetic, applying OPEN-SQUARES-MOVE-REWRITE, ENDP-OPEN-SQUARES, LEGALP-PICK-MOVE, LENGTH-MOVE, LESSP-OPEN-SQUARES-LENGTH, ADD1-DIFFERENCE, and CAR-CONS, and expanding the functions LEGALP, EQUAL, LESSP, TIC-TAC-TOE1, TIC-TAC-TOE1-A, and ADD1, to two new conjectures: Case 2.2. (IMPLIES (AND (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9) (NOT (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9))) (EQUAL (CONS (MOVE 'X (CAR XMOVES) BOARD) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))))) (CONS (MOVE 'X (CAR XMOVES) BOARD) '(DRAW)))), which again simplifies, rewriting with the lemmas CAR-CONS, CDR-CONS, and CONS-EQUAL, and unfolding CAR, to: (IMPLIES (AND (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). Appealing to the lemma CAR-CDR-ELIM, we now replace XMOVES by (CONS X Z) to eliminate (CAR XMOVES) and (CDR XMOVES). This generates two new goals: Case 2.2.2. (IMPLIES (AND (NOT (LISTP XMOVES)) (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)), which further simplifies, applying CAR-NLISTP, CDR-NLISTP, CAR-CONS, and CDR-CONS, and opening up LESSP, EQUAL, NTH, MOVE, LEGALP, CAR, TIC-TAC-TOE1, OPEN-SQUARES, SUB1, DIFFERENCE, TIC-TAC-TOE1-A, and LENGTH, to the following three new conjectures: Case 2.2.2.3. (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (CONS 'X (CDR BOARD))) (CONS 'X (CDR BOARD))))) (NOT (EQUAL (PICK-MOVE (CONS 'X (CDR BOARD))) 0)) (NUMBERP (PICK-MOVE (CONS 'X (CDR BOARD)))) (EQUAL (TIC-TAC-TOE1 0 (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD)))) (TIC-TAC-TOE1-A 0 (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD))) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). This again simplifies, applying CDR-CONS and CAR-CONS, and expanding MOVE, LEGALP, LESSP, EQUAL, NTH, CAR, TIC-TAC-TOE1, and TIC-TAC-TOE1-A, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (NOT (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X (CDR BOARD)))) (CDR BOARD))))) (NOT (EQUAL (PICK-MOVE (CONS 'X (CDR BOARD))) 0)) (NUMBERP (PICK-MOVE (CONS 'X (CDR BOARD)))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). Applying the lemma CAR-CDR-ELIM, replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). We would thus like to prove: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT X) (NOT (EQUAL (SUB1 (OPEN-SQUARES (CONS X Z))) 0)) (NOT (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)))) (NOT (EQUAL (PICK-MOVE (CONS 'X Z)) 0)) (NUMBERP (PICK-MOVE (CONS 'X Z))) (EQUAL (LENGTH (CONS X Z)) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES (CONS X Z))) 9)), which further simplifies, applying CDR-CONS, CAR-CONS, SUB1-ADD1, and ADD1-EQUAL, and opening up the functions OPEN-SQUARES, LENGTH, NUMBERP, SUB1, EQUAL, and DIFFERENCE, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)))) (NOT (EQUAL (PICK-MOVE (CONS 'X Z)) 0)) (NUMBERP (PICK-MOVE (CONS 'X Z))) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)), which again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP 9 (OPEN-SQUARES Z)) (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (WINP 'O (CONS 'X (MOVE 'O (SUB1 (PICK-MOVE (CONS 'X Z))) Z)))) (NOT (EQUAL (PICK-MOVE (CONS 'X Z)) 0)) (NUMBERP (PICK-MOVE (CONS 'X Z))) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)). This finally simplifies, using linear arithmetic and appealing to the lemma LESSP-OPEN-SQUARES-LENGTH, to: T. Case 2.2.2.2. (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (CONS 'X (CDR BOARD))) (CONS 'X (CDR BOARD))))) (EQUAL (PICK-MOVE (CONS 'X (CDR BOARD))) 0) (EQUAL (TIC-TAC-TOE1 0 (CONS 'O (CDR BOARD))) (TIC-TAC-TOE1-A 0 (CONS 'O (CDR BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)), which again simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definitions of EQUAL, MOVE, LEGALP, LESSP, NTH, CAR, TIC-TAC-TOE1, and TIC-TAC-TOE1-A, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (NOT (WINP 'O (CONS 'O (CDR BOARD)))) (EQUAL (PICK-MOVE (CONS 'X (CDR BOARD))) 0) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). Applying the lemma CAR-CDR-ELIM, replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). We thus obtain the new conjecture: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT X) (NOT (EQUAL (SUB1 (OPEN-SQUARES (CONS X Z))) 0)) (NOT (WINP 'O (CONS 'O Z))) (EQUAL (PICK-MOVE (CONS 'X Z)) 0) (EQUAL (LENGTH (CONS X Z)) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES (CONS X Z))) 9)), which further simplifies, applying the lemmas CDR-CONS, CAR-CONS, SUB1-ADD1, and ADD1-EQUAL, and expanding the functions OPEN-SQUARES, LENGTH, NUMBERP, SUB1, EQUAL, and DIFFERENCE, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (WINP 'O (CONS 'O Z))) (EQUAL (PICK-MOVE (CONS 'X Z)) 0) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)). This again simplifies, using linear arithmetic, to the goal: (IMPLIES (AND (LESSP 9 (OPEN-SQUARES Z)) (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (WINP 'O (CONS 'O Z))) (EQUAL (PICK-MOVE (CONS 'X Z)) 0) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)). However this finally simplifies, using linear arithmetic and rewriting with LESSP-OPEN-SQUARES-LENGTH, to: T. Case 2.2.2.1. (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (CONS 'X (CDR BOARD))) (CONS 'X (CDR BOARD))))) (NOT (NUMBERP (PICK-MOVE (CONS 'X (CDR BOARD))))) (EQUAL (TIC-TAC-TOE1 0 (CONS 'O (CDR BOARD))) (TIC-TAC-TOE1-A 0 (CONS 'O (CDR BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). But this again simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions MOVE, LEGALP, LESSP, EQUAL, NTH, CAR, TIC-TAC-TOE1, and TIC-TAC-TOE1-A, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (CAR BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (LISTP BOARD) (NOT (WINP 'O (CONS 'O (CDR BOARD)))) (NOT (NUMBERP (PICK-MOVE (CONS 'X (CDR BOARD))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). Applying the lemma CAR-CDR-ELIM, replace BOARD by (CONS X Z) to eliminate (CAR BOARD) and (CDR BOARD). We would thus like to prove: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT X) (NOT (EQUAL (SUB1 (OPEN-SQUARES (CONS X Z))) 0)) (NOT (WINP 'O (CONS 'O Z))) (NOT (NUMBERP (PICK-MOVE (CONS 'X Z)))) (EQUAL (LENGTH (CONS X Z)) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES (CONS X Z))) 9)), which further simplifies, applying CDR-CONS, CAR-CONS, SUB1-ADD1, and ADD1-EQUAL, and opening up the definitions of OPEN-SQUARES, LENGTH, NUMBERP, SUB1, EQUAL, and DIFFERENCE, to: (IMPLIES (AND (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (WINP 'O (CONS 'O Z))) (NOT (NUMBERP (PICK-MOVE (CONS 'X Z)))) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)), which again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP 9 (OPEN-SQUARES Z)) (NOT (LISTP XMOVES)) (NOT (EQUAL (OPEN-SQUARES Z) 0)) (NOT (WINP 'O (CONS 'O Z))) (NOT (NUMBERP (PICK-MOVE (CONS 'X Z)))) (EQUAL (LENGTH Z) 8)) (LESSP (DIFFERENCE 9 (OPEN-SQUARES Z)) 9)). But this finally simplifies, using linear arithmetic and applying LESSP-OPEN-SQUARES-LENGTH, to: T. Case 2.2.1. (IMPLIES (AND (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)))) (EQUAL (TIC-TAC-TOE1 Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)). We will try to prove the above formula by generalizing it, replacing (OPEN-SQUARES BOARD) by Y and (MOVE (QUOTE X) X BOARD) by A. We restrict the new variables by recalling the type restriction lemma noted when OPEN-SQUARES was introduced. We would thus like to prove: (IMPLIES (AND (NUMBERP Y) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 Y) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE A) A))) (EQUAL (TIC-TAC-TOE1 Z (MOVE 'O (PICK-MOVE A) A)) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 10 (SUB1 (SUB1 Y))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 Y) 9)). Applying the lemma SUB1-ELIM, replace Y by (ADD1 V) to eliminate (SUB1 Y) and V by (ADD1 W) to eliminate (SUB1 V). We rely upon the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove the following two new goals: Case 2.2.1.2. (IMPLIES (AND (EQUAL Y 0) (NUMBERP Y) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 Y) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE A) A))) (EQUAL (TIC-TAC-TOE1 Z (MOVE 'O (PICK-MOVE A) A)) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 10 (SUB1 (SUB1 Y))))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 Y) 9)). This finally simplifies, expanding the functions NUMBERP, SUB1, and EQUAL, to: T. Case 2.2.1.1. (IMPLIES (AND (NUMBERP W) (NOT (EQUAL (ADD1 (ADD1 W)) 0)) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (ADD1 W) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE A) A))) (EQUAL (TIC-TAC-TOE1 Z (MOVE 'O (PICK-MOVE A) A)) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 10 W))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (ADD1 (ADD1 W))) 9)), which further simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP 10 (ADD1 (ADD1 W))) (NUMBERP W) (NOT (EQUAL (ADD1 (ADD1 W)) 0)) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (ADD1 W) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE A) A))) (EQUAL (TIC-TAC-TOE1 Z (MOVE 'O (PICK-MOVE A) A)) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 10 W))) (EQUAL (LENGTH BOARD) 9)) (LESSP (DIFFERENCE 10 (ADD1 (ADD1 W))) 9)). But this finally simplifies, using linear arithmetic, applying SUB1-ADD1 and DIFFERENCE-IS-0, and unfolding the definitions of SUB1, NUMBERP, EQUAL, and LESSP, to: T. Case 2.1. (IMPLIES (AND (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)) (EQUAL (CONS (MOVE 'X (CAR XMOVES) BOARD) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))))) (CONS (MOVE 'X (CAR XMOVES) BOARD) (CONS (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 12 (OPEN-SQUARES BOARD))))))). This again simplifies, rewriting with CAR-CONS, CDR-CONS, and CONS-EQUAL, to: (IMPLIES (AND (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 12 (OPEN-SQUARES BOARD))))). Applying the lemma CAR-CDR-ELIM, replace XMOVES by (CONS X Z) to eliminate (CAR XMOVES) and (CDR XMOVES). This produces the following two new formulas: Case 2.1.2. (IMPLIES (AND (NOT (LISTP XMOVES)) (LESSP (CAR XMOVES) 9) (NOT (NTH (CAR XMOVES) BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)) (EQUAL (TIC-TAC-TOE1 (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (DIFFERENCE 12 (OPEN-SQUARES BOARD))))). However this further simplifies, rewriting with CAR-NLISTP, CDR-NLISTP, CAR-CONS, and CDR-CONS, and opening up the definitions of LESSP, EQUAL, NTH, MOVE, LEGALP, CAR, TIC-TAC-TOE1, OPEN-SQUARES, SUB1, DIFFERENCE, TIC-TAC-TOE1-A, and LENGTH, to: T. Case 2.1.1. (IMPLIES (AND (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)))) (EQUAL (TIC-TAC-TOE1 Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD)))))) (EQUAL (LENGTH BOARD) 9) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)) (EQUAL (TIC-TAC-TOE1 Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)) (DIFFERENCE 12 (OPEN-SQUARES BOARD))))). We use the first equality hypothesis by substituting: (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD))))) for: (TIC-TAC-TOE1 Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD))) and throwing away the equality. This produces: (IMPLIES (AND (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 (OPEN-SQUARES BOARD)) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)))) (EQUAL (LENGTH BOARD) 9) (LESSP (DIFFERENCE 10 (OPEN-SQUARES BOARD)) 9)) (EQUAL (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)) (DIFFERENCE 10 (SUB1 (SUB1 (OPEN-SQUARES BOARD))))) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE (MOVE 'X X BOARD)) (MOVE 'X X BOARD)) (DIFFERENCE 12 (OPEN-SQUARES BOARD))))), which we generalize by replacing (OPEN-SQUARES BOARD) by Y and (MOVE (QUOTE X) X BOARD) by A. We restrict the new variables by recalling the type restriction lemma noted when OPEN-SQUARES was introduced. We thus obtain: (IMPLIES (AND (NUMBERP Y) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 Y) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE A) A))) (EQUAL (LENGTH BOARD) 9) (LESSP (DIFFERENCE 10 Y) 9)) (EQUAL (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 10 (SUB1 (SUB1 Y)))) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 12 Y)))). Applying the lemma SUB1-ELIM, replace Y by (ADD1 V) to eliminate (SUB1 Y) and V by (ADD1 W) to eliminate (SUB1 V). We use the type restriction lemma noted when SUB1 was introduced to restrict the new variable. We would thus like to prove the following two new goals: Case 2.1.1.2. (IMPLIES (AND (EQUAL Y 0) (NUMBERP Y) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (SUB1 Y) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE A) A))) (EQUAL (LENGTH BOARD) 9) (LESSP (DIFFERENCE 10 Y) 9)) (EQUAL (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 10 (SUB1 (SUB1 Y)))) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 12 Y)))). But this finally simplifies, using linear arithmetic, to: T. Case 2.1.1.1. (IMPLIES (AND (NUMBERP W) (NOT (EQUAL (ADD1 (ADD1 W)) 0)) (LESSP X 9) (NOT (NTH X BOARD)) (NOT (EQUAL (ADD1 W) 0)) (NOT (WINP 'O (MOVE 'O (PICK-MOVE A) A))) (EQUAL (LENGTH BOARD) 9) (LESSP (DIFFERENCE 10 (ADD1 (ADD1 W))) 9)) (EQUAL (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 10 W)) (TIC-TAC-TOE1-A Z (MOVE 'O (PICK-MOVE A) A) (DIFFERENCE 12 (ADD1 (ADD1 W)))))), which finally simplifies, rewriting with SUB1-ADD1, and opening up SUB1, NUMBERP, EQUAL, and DIFFERENCE, to: T. Case 1. (IMPLIES (AND (NOT (LEGALP (CAR XMOVES) BOARD)) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1 XMOVES BOARD) (TIC-TAC-TOE1-A XMOVES BOARD (DIFFERENCE 10 (OPEN-SQUARES BOARD))))). This simplifies, expanding the functions LEGALP, TIC-TAC-TOE1, and TIC-TAC-TOE1-A, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 10.9 0.0 ] LEMMA-A1 (DISABLE ADD1-DIFFERENCE) [ 0.0 0.0 0.0 ] ADD1-DIFFERENCE-OFF (PROVE-LEMMA LEMMA-A (REWRITE) (EQUAL (TIC-TAC-TOE XMOVES) (TIC-TAC-TOE-A XMOVES)) ((USE (LEMMA-A1 (BOARD (LIST F F F F F F F F F)))))) WARNING: Note that the rewrite rule LEMMA-A will be stored so as to apply only to terms with the nonrecursive function symbol TIC-TAC-TOE. This conjecture can be simplified, using the abbreviations TIC-TAC-TOE-A and TIC-TAC-TOE, to: (IMPLIES (IMPLIES (EQUAL (LENGTH (LIST F F F F F F F F F)) 9) (EQUAL (TIC-TAC-TOE1 XMOVES (LIST F F F F F F F F F)) (TIC-TAC-TOE1-A XMOVES (LIST F F F F F F F F F) (DIFFERENCE 10 (OPEN-SQUARES (LIST F F F F F F F F F)))))) (EQUAL (CONS (LIST F F F F F F F F F) (TIC-TAC-TOE1 XMOVES (LIST F F F F F F F F F))) (CONS (LIST F F F F F F F F F) (TIC-TAC-TOE1-A XMOVES (LIST F F F F F F F F F) 1)))). This simplifies, unfolding the definitions of LENGTH, EQUAL, OPEN-SQUARES, DIFFERENCE, and IMPLIES, to: T. Q.E.D. [ 0.0 2.0 0.0 ] LEMMA-A (DEFN PIECE->NUMBER (X) (IF (EQUAL X F) 0 (IF (EQUAL X 'X) 1 2))) Observe that (NUMBERP (PIECE->NUMBER X)) is a theorem. [ 0.0 0.0 0.0 ] PIECE->NUMBER (DEFN NUMBER->PIECE (J) (IF (EQUAL J 0) F (IF (EQUAL J 1) 'X 'O))) Observe that (OR (FALSEP (NUMBER->PIECE J)) (LITATOM (NUMBER->PIECE J))) is a theorem. [ 0.0 0.0 0.0 ] NUMBER->PIECE (DEFN BOARD->NUMBER (BOARD) (IF (NLISTP BOARD) 0 (PLUS (PIECE->NUMBER (CAR BOARD)) (TIMES 4 (BOARD->NUMBER (CDR BOARD)))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Hence, BOARD->NUMBER is accepted under the principle of definition. From the definition we can conclude that (NUMBERP (BOARD->NUMBER BOARD)) is a theorem. [ 0.0 0.0 0.0 ] BOARD->NUMBER (DEFN NUMBER->BOARD (N LEN) (IF (ZEROP LEN) NIL (CONS (NUMBER->PIECE (REMAINDER N 4)) (NUMBER->BOARD (QUOTIENT N 4) (SUB1 LEN))))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to establish that the measure (COUNT LEN) decreases according to the well-founded relation LESSP in each recursive call. Hence, NUMBER->BOARD is accepted under the principle of definition. Note that: (OR (LITATOM (NUMBER->BOARD N LEN)) (LISTP (NUMBER->BOARD N LEN))) is a theorem. [ 0.0 0.0 0.0 ] NUMBER->BOARD (DEFN BOARDP* (LST LEN) (IF (NLISTP LST) (EQUAL LST NIL) (AND (BOARDP (CAR LST)) (EQUAL (LENGTH (CAR LST)) LEN) (BOARDP* (CDR LST) LEN)))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to show that the measure (COUNT LST) decreases according to the well-founded relation LESSP in each recursive call. Hence, BOARDP* is accepted under the definitional principle. Note that: (OR (FALSEP (BOARDP* LST LEN)) (TRUEP (BOARDP* LST LEN))) is a theorem. [ 0.0 0.0 0.0 ] BOARDP* (DEFN NUMBER->BOARD* (LST LEN) (IF (NLISTP LST) NIL (CONS (IF (EQUAL (CAR LST) -2) 'O-WIN (IF (EQUAL (CAR LST) -1) 'DRAW (NUMBER->BOARD (CAR LST) LEN))) (NUMBER->BOARD* (CDR LST) LEN)))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to establish that the measure (COUNT LST) decreases according to the well-founded relation LESSP in each recursive call. Hence, NUMBER->BOARD* is accepted under the definitional principle. Note that: (OR (LITATOM (NUMBER->BOARD* LST LEN)) (LISTP (NUMBER->BOARD* LST LEN))) is a theorem. [ 0.0 0.0 0.0 ] NUMBER->BOARD* (PROVE-LEMMA REMAINDER-PLUS-TIMES (REWRITE) (IMPLIES (AND (NUMBERP X) (LESSP X Y)) (EQUAL (REMAINDER (PLUS X (TIMES Y Z)) Y) X)) ((INDUCT (TIMES Z Y)))) This formula can be simplified, using the abbreviations ZEROP, IMPLIES, NOT, OR, and AND, to the following two new goals: Case 2. (IMPLIES (AND (ZEROP Z) (NUMBERP X) (LESSP X Y)) (EQUAL (REMAINDER (PLUS X (TIMES Y Z)) Y) X)). This simplifies, rewriting with TIMES-COMMUTES1, PLUS-COMMUTES1, and TIMES-NON-NUMBERP, and opening up the functions ZEROP, EQUAL, TIMES, PLUS, and REMAINDER, to: T. Case 1. (IMPLIES (AND (NOT (EQUAL Z 0)) (NUMBERP Z) (IMPLIES (AND (NUMBERP X) (LESSP X Y)) (EQUAL (REMAINDER (PLUS X (TIMES Y (SUB1 Z))) Y) X)) (NUMBERP X) (LESSP X Y)) (EQUAL (REMAINDER (PLUS X (TIMES Y Z)) Y) X)), which simplifies, opening up AND and IMPLIES, to: (IMPLIES (AND (NOT (EQUAL Z 0)) (NUMBERP Z) (EQUAL (REMAINDER (PLUS X (TIMES Y (SUB1 Z))) Y) X) (LESSP X Y)) (EQUAL (REMAINDER (PLUS X (TIMES Y Z)) Y) X)). Appealing to the lemma SUB1-ELIM, we now replace Z by (ADD1 V) to eliminate (SUB1 Z). We use the type restriction lemma noted when SUB1 was introduced to constrain the new variable. We must thus prove: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (EQUAL (REMAINDER (PLUS X (TIMES Y V)) Y) X) (LESSP X Y)) (EQUAL (REMAINDER (PLUS X (TIMES Y (ADD1 V))) Y) X)). This further simplifies, applying TIMES-COMMUTES1, TIMES-ADD1, and DIFFERENCE-PLUS-CANCELLATION3, and opening up the definition of REMAINDER, to the following three new formulas: Case 1.3. (IMPLIES (AND (NUMBERP V) (EQUAL (REMAINDER (PLUS X (TIMES V Y)) Y) X) (LESSP X Y) (NOT (NUMBERP Y))) (EQUAL (PLUS X Y (TIMES V Y)) X)). This again simplifies, appealing to the lemmas TIMES-NON-NUMBERP and PLUS-COMMUTES1, and expanding the functions EQUAL, PLUS, REMAINDER, and LESSP, to: T. Case 1.2. (IMPLIES (AND (NUMBERP V) (EQUAL (REMAINDER (PLUS X (TIMES V Y)) Y) X) (LESSP X Y) (EQUAL Y 0)) (EQUAL (PLUS X Y (TIMES V Y)) X)), which again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (NUMBERP V) (EQUAL (REMAINDER (PLUS X (TIMES V Y)) Y) X) (LESSP X Y) (LESSP (PLUS X Y (TIMES V Y)) Y)) (EQUAL (PLUS X Y (TIMES V Y)) X)), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] REMAINDER-PLUS-TIMES (PROVE-LEMMA REMAINDER-PLUS-TIMES-INSTANCE (REWRITE) (IMPLIES (NOT (ZEROP Y)) (EQUAL (REMAINDER (TIMES Y Z) Y) 0)) ((USE (REMAINDER-PLUS-TIMES (X 0))) (DISABLE REMAINDER-PLUS-TIMES))) This formula can be simplified, using the abbreviations ZEROP, NOT, and IMPLIES, to: (IMPLIES (AND (IMPLIES (AND (NUMBERP 0) (LESSP 0 Y)) (EQUAL (REMAINDER (PLUS 0 (TIMES Y Z)) Y) 0)) (NOT (EQUAL Y 0)) (NUMBERP Y)) (EQUAL (REMAINDER (TIMES Y Z) Y) 0)), which simplifies, unfolding the functions NUMBERP, EQUAL, LESSP, AND, PLUS, and IMPLIES, to: T. Q.E.D. [ 0.0 0.0 0.0 ] REMAINDER-PLUS-TIMES-INSTANCE (PROVE-LEMMA QUOTIENT-PLUS-TIMES (REWRITE) (IMPLIES (AND (NUMBERP Z) (LESSP X Y)) (EQUAL (QUOTIENT (PLUS X (TIMES Y Z)) Y) Z)) ((INDUCT (TIMES Z Y)))) This formula can be simplified, using the abbreviations ZEROP, IMPLIES, NOT, OR, and AND, to the following two new conjectures: Case 2. (IMPLIES (AND (ZEROP Z) (NUMBERP Z) (LESSP X Y)) (EQUAL (QUOTIENT (PLUS X (TIMES Y Z)) Y) Z)). This simplifies, applying TIMES-COMMUTES1 and PLUS-COMMUTES1, and expanding the definitions of ZEROP, NUMBERP, EQUAL, TIMES, and PLUS, to two new conjectures: Case 2.2. (IMPLIES (AND (EQUAL Z 0) (LESSP X Y) (NOT (NUMBERP X))) (EQUAL (QUOTIENT 0 Y) 0)), which again simplifies, opening up LESSP, EQUAL, and QUOTIENT, to: T. Case 2.1. (IMPLIES (AND (EQUAL Z 0) (LESSP X Y) (NUMBERP X)) (EQUAL (QUOTIENT X Y) 0)), which again simplifies, expanding QUOTIENT and EQUAL, to: T. Case 1. (IMPLIES (AND (NOT (EQUAL Z 0)) (NUMBERP Z) (IMPLIES (AND (NUMBERP (SUB1 Z)) (LESSP X Y)) (EQUAL (QUOTIENT (PLUS X (TIMES Y (SUB1 Z))) Y) (SUB1 Z))) (LESSP X Y)) (EQUAL (QUOTIENT (PLUS X (TIMES Y Z)) Y) Z)), which simplifies, opening up the functions AND and IMPLIES, to the formula: (IMPLIES (AND (NOT (EQUAL Z 0)) (NUMBERP Z) (EQUAL (QUOTIENT (PLUS X (TIMES Y (SUB1 Z))) Y) (SUB1 Z)) (LESSP X Y)) (EQUAL (QUOTIENT (PLUS X (TIMES Y Z)) Y) Z)). Appealing to the lemma SUB1-ELIM, we now replace Z by (ADD1 V) to eliminate (SUB1 Z). We use the type restriction lemma noted when SUB1 was introduced to constrain the new variable. We must thus prove: (IMPLIES (AND (NUMBERP V) (NOT (EQUAL (ADD1 V) 0)) (EQUAL (QUOTIENT (PLUS X (TIMES Y V)) Y) V) (LESSP X Y)) (EQUAL (QUOTIENT (PLUS X (TIMES Y (ADD1 V))) Y) (ADD1 V))). However this further simplifies, rewriting with TIMES-COMMUTES1, TIMES-ADD1, and DIFFERENCE-PLUS-CANCELLATION3, and expanding the definition of QUOTIENT, to the following three new formulas: Case 1.3. (IMPLIES (AND (EQUAL (QUOTIENT (PLUS X (TIMES V Y)) Y) V) (LESSP X Y) (NOT (NUMBERP Y))) (EQUAL 0 (ADD1 V))). This again simplifies, rewriting with the lemmas TIMES-NON-NUMBERP and PLUS-COMMUTES1, and opening up the functions EQUAL, PLUS, QUOTIENT, and LESSP, to: T. Case 1.2. (IMPLIES (AND (EQUAL (QUOTIENT (PLUS X (TIMES V Y)) Y) V) (LESSP X Y) (EQUAL Y 0)) (EQUAL 0 (ADD1 V))), which again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (EQUAL (QUOTIENT (PLUS X (TIMES V Y)) Y) V) (LESSP X Y) (LESSP (PLUS X Y (TIMES V Y)) Y)) (EQUAL 0 (ADD1 V))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] QUOTIENT-PLUS-TIMES (PROVE-LEMMA QUOTIENT-PLUS-TIMES-INSTANCE (REWRITE) (IMPLIES (AND (NUMBERP Z) (NOT (ZEROP Y))) (EQUAL (QUOTIENT (TIMES Y Z) Y) Z)) ((USE (QUOTIENT-PLUS-TIMES (X 0))) (DISABLE QUOTIENT-PLUS-TIMES))) This conjecture can be simplified, using the abbreviations ZEROP, NOT, AND, and IMPLIES, to: (IMPLIES (AND (IMPLIES (AND (NUMBERP Z) (LESSP 0 Y)) (EQUAL (QUOTIENT (PLUS 0 (TIMES Y Z)) Y) Z)) (NUMBERP Z) (NOT (EQUAL Y 0)) (NUMBERP Y)) (EQUAL (QUOTIENT (TIMES Y Z) Y) Z)). This simplifies, expanding the definitions of EQUAL, LESSP, AND, PLUS, and IMPLIES, to: T. Q.E.D. [ 0.0 0.0 0.0 ] QUOTIENT-PLUS-TIMES-INSTANCE (PROVE-LEMMA BOARD->NUMBER->BOARD (REWRITE) (IMPLIES (BOARDP BOARD) (EQUAL (NUMBER->BOARD (BOARD->NUMBER BOARD) (LENGTH BOARD)) BOARD))) Call the conjecture *1. Perhaps we can prove it by induction. Three inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (p (CDR BOARD))) (p BOARD))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to prove that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to three new goals: Case 3. (IMPLIES (AND (NLISTP BOARD) (BOARDP BOARD)) (EQUAL (NUMBER->BOARD (BOARD->NUMBER BOARD) (LENGTH BOARD)) BOARD)), which simplifies, opening up the definitions of NLISTP, BOARDP, BOARD->NUMBER, LENGTH, NUMBER->BOARD, and EQUAL, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (BOARDP (CDR BOARD))) (BOARDP BOARD)) (EQUAL (NUMBER->BOARD (BOARD->NUMBER BOARD) (LENGTH BOARD)) BOARD)), which simplifies, expanding NLISTP and BOARDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (EQUAL (NUMBER->BOARD (BOARD->NUMBER (CDR BOARD)) (LENGTH (CDR BOARD))) (CDR BOARD)) (BOARDP BOARD)) (EQUAL (NUMBER->BOARD (BOARD->NUMBER BOARD) (LENGTH BOARD)) BOARD)), which simplifies, applying the lemmas SUB1-ADD1, QUOTIENT-PLUS-TIMES, REMAINDER-PLUS-TIMES, CAR-CONS, CDR-CONS, QUOTIENT-PLUS-TIMES-INSTANCE, and REMAINDER-PLUS-TIMES-INSTANCE, and opening up the functions NLISTP, BOARDP, BOARD->NUMBER, PIECE->NUMBER, LENGTH, NUMBER->PIECE, LESSP, NUMBER->BOARD, EQUAL, and PLUS, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] BOARD->NUMBER->BOARD (DEFN MOVEN (X I BOARD) (COND ((ZEROP I) (PLUS X (TIMES 4 (QUOTIENT BOARD 4)))) (T (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN X (SUB1 I) (QUOTIENT BOARD 4))))))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to show that the measure (COUNT I) decreases according to the well-founded relation LESSP in each recursive call. Hence, MOVEN is accepted under the principle of definition. From the definition we can conclude that: (NUMBERP (MOVEN X I BOARD)) is a theorem. [ 0.0 0.0 0.0 ] MOVEN (PROVE-LEMMA MOVE-IS-MOVEN (REWRITE) (IMPLIES (AND (OR (EQUAL X 'X) (EQUAL X 'O)) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD))))) This conjecture simplifies, unfolding the functions OR and PIECE->NUMBER, to the following two new conjectures: Case 2. (IMPLIES (AND (EQUAL X 'X) (BOARDP BOARD)) (EQUAL (MOVE 'X I BOARD) (NUMBER->BOARD (MOVEN 1 I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))). This again simplifies, clearly, to: (IMPLIES (BOARDP BOARD) (EQUAL (MOVE 'X I BOARD) (NUMBER->BOARD (MOVEN 1 I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))), which we will name *1. Case 1. (IMPLIES (AND (EQUAL X 'O) (BOARDP BOARD)) (EQUAL (MOVE 'O I BOARD) (NUMBER->BOARD (MOVEN 2 I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))). This again simplifies, trivially, to: (IMPLIES (BOARDP BOARD) (EQUAL (MOVE 'O I BOARD) (NUMBER->BOARD (MOVEN 2 I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (AND (IMPLIES (AND (EQUAL X 'X) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))) (IMPLIES (AND (EQUAL X 'O) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD))))). We named this *1. We will try to prove it by induction. 12 inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p X I BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I)) (p X I BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (p X (SUB1 I) (CDR BOARD))) (p X I BOARD))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definitions of ZEROP and NLISTP inform us that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme generates eight new formulas: Case 8. (IMPLIES (AND (NLISTP BOARD) (EQUAL X 'X) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))), which simplifies, expanding the functions NLISTP, BOARDP, LISTP, MOVE, PIECE->NUMBER, BOARD->NUMBER, LENGTH, EQUAL, and NUMBER->BOARD, to: T. Case 7. (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I) (EQUAL X 'X) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))), which simplifies, appealing to the lemmas QUOTIENT-PLUS-TIMES, BOARD->NUMBER->BOARD, SUB1-ADD1, REMAINDER-PLUS-TIMES, and QUOTIENT-PLUS-TIMES-INSTANCE, and opening up the definitions of NLISTP, ZEROP, BOARDP, EQUAL, MOVE, PIECE->NUMBER, BOARD->NUMBER, LESSP, MOVEN, LENGTH, NUMBER->PIECE, NUMBER->BOARD, and PLUS, to: T. Case 6. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (EQUAL (MOVE X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL X 'X) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))), which simplifies, rewriting with SUB1-ADD1, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of NLISTP, ZEROP, PIECE->NUMBER, BOARDP, MOVE, BOARD->NUMBER, LENGTH, NUMBER->PIECE, NUMBER->BOARD, EQUAL, and PLUS, to the following eight new conjectures: Case 6.8. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 1 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD))) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (QUOTIENT (MOVEN 1 I (PLUS 2 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) (LENGTH (CDR BOARD))))). However this again simplifies, applying QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and unfolding LESSP and MOVEN, to: T. Case 6.7. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 1 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD))) (NOT (EQUAL (REMAINDER (MOVEN 1 I (PLUS 2 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) 1))). But this again simplifies, applying QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and opening up LESSP, MOVEN, and EQUAL, to: T. Case 6.6. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 1 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD))) (NOT (EQUAL (REMAINDER (MOVEN 1 I (PLUS 2 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) 0))). This again simplifies, applying the lemmas QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and opening up the functions LESSP, MOVEN, and EQUAL, to: T. Case 6.5. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 1 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD))) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (QUOTIENT (MOVEN 1 I (PLUS 1 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) (LENGTH (CDR BOARD))))), which again simplifies, rewriting with QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and unfolding the functions LESSP and MOVEN, to: T. Case 6.4. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 1 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD))) (EQUAL (REMAINDER (MOVEN 1 I (PLUS 1 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) 1)). However this again simplifies, rewriting with the lemmas QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and expanding the functions LESSP, MOVEN, and EQUAL, to: T. Case 6.3. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 1 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD))) (NOT (EQUAL (REMAINDER (MOVEN 1 I (PLUS 1 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) 0))), which again simplifies, rewriting with QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and expanding LESSP, MOVEN, and EQUAL, to: T. Case 6.2. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 1 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (NOT (CAR BOARD)) (BOARDP (CDR BOARD))) (EQUAL (REMAINDER (MOVEN 1 I (TIMES 4 (BOARD->NUMBER (CDR BOARD)))) 4) 0)). But this again simplifies, applying QUOTIENT-PLUS-TIMES-INSTANCE and REMAINDER-PLUS-TIMES-INSTANCE, and opening up PLUS, EQUAL, and MOVEN, to: T. Case 6.1. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 1 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (NOT (CAR BOARD)) (BOARDP (CDR BOARD))) (EQUAL (MOVE 'X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (QUOTIENT (MOVEN 1 I (TIMES 4 (BOARD->NUMBER (CDR BOARD)))) 4) (LENGTH (CDR BOARD))))). This again simplifies, rewriting with QUOTIENT-PLUS-TIMES-INSTANCE and REMAINDER-PLUS-TIMES-INSTANCE, and expanding the functions PLUS, EQUAL, and MOVEN, to: T. Case 5. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (BOARDP (CDR BOARD))) (EQUAL X 'X) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))). This simplifies, unfolding the definitions of NLISTP, ZEROP, and BOARDP, to: T. Case 4. (IMPLIES (AND (NLISTP BOARD) (EQUAL X 'O) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))). This simplifies, opening up the functions NLISTP, BOARDP, LISTP, MOVE, PIECE->NUMBER, BOARD->NUMBER, LENGTH, EQUAL, and NUMBER->BOARD, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (ZEROP I) (EQUAL X 'O) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))). This simplifies, rewriting with QUOTIENT-PLUS-TIMES, BOARD->NUMBER->BOARD, SUB1-ADD1, REMAINDER-PLUS-TIMES, and QUOTIENT-PLUS-TIMES-INSTANCE, and opening up the functions NLISTP, ZEROP, BOARDP, EQUAL, MOVE, PIECE->NUMBER, BOARD->NUMBER, LESSP, MOVEN, LENGTH, NUMBER->PIECE, NUMBER->BOARD, and PLUS, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (EQUAL (MOVE X (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL X 'O) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))), which simplifies, rewriting with SUB1-ADD1, CAR-CONS, and CONS-EQUAL, and unfolding NLISTP, ZEROP, PIECE->NUMBER, BOARDP, MOVE, BOARD->NUMBER, LENGTH, NUMBER->PIECE, NUMBER->BOARD, EQUAL, and PLUS, to the following eight new goals: Case 2.8. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 2 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD))) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (QUOTIENT (MOVEN 2 I (PLUS 2 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) (LENGTH (CDR BOARD))))). But this again simplifies, appealing to the lemmas QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and opening up the definitions of LESSP and MOVEN, to: T. Case 2.7. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 2 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD))) (NOT (EQUAL (REMAINDER (MOVEN 2 I (PLUS 2 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) 1))), which again simplifies, rewriting with the lemmas QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and unfolding the functions LESSP, MOVEN, and EQUAL, to: T. Case 2.6. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 2 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'O) (BOARDP (CDR BOARD))) (NOT (EQUAL (REMAINDER (MOVEN 2 I (PLUS 2 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) 0))), which again simplifies, applying QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and unfolding the functions LESSP, MOVEN, and EQUAL, to: T. Case 2.5. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 2 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD))) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (QUOTIENT (MOVEN 2 I (PLUS 1 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) (LENGTH (CDR BOARD))))). But this again simplifies, applying QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and opening up the functions LESSP and MOVEN, to: T. Case 2.4. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 2 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD))) (EQUAL (REMAINDER (MOVEN 2 I (PLUS 1 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) 1)). But this again simplifies, applying the lemmas QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and unfolding the functions LESSP, MOVEN, and EQUAL, to: T. Case 2.3. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 2 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (EQUAL (CAR BOARD) 'X) (BOARDP (CDR BOARD))) (NOT (EQUAL (REMAINDER (MOVEN 2 I (PLUS 1 (TIMES 4 (BOARD->NUMBER (CDR BOARD))))) 4) 0))), which again simplifies, applying QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and expanding the definitions of LESSP, MOVEN, and EQUAL, to: T. Case 2.2. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 2 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (NOT (CAR BOARD)) (BOARDP (CDR BOARD))) (EQUAL (REMAINDER (MOVEN 2 I (TIMES 4 (BOARD->NUMBER (CDR BOARD)))) 4) 0)). But this again simplifies, rewriting with QUOTIENT-PLUS-TIMES-INSTANCE and REMAINDER-PLUS-TIMES-INSTANCE, and opening up the definitions of PLUS, EQUAL, and MOVEN, to: T. Case 2.1. (IMPLIES (AND (LISTP BOARD) (NOT (EQUAL I 0)) (NUMBERP I) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (MOVEN 2 (SUB1 I) (BOARD->NUMBER (CDR BOARD))) (LENGTH (CDR BOARD)))) (NOT (CAR BOARD)) (BOARDP (CDR BOARD))) (EQUAL (MOVE 'O (SUB1 I) (CDR BOARD)) (NUMBER->BOARD (QUOTIENT (MOVEN 2 I (TIMES 4 (BOARD->NUMBER (CDR BOARD)))) 4) (LENGTH (CDR BOARD))))). This again simplifies, rewriting with QUOTIENT-PLUS-TIMES-INSTANCE and REMAINDER-PLUS-TIMES-INSTANCE, and unfolding the functions PLUS, EQUAL, and MOVEN, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (ZEROP I)) (NOT (BOARDP (CDR BOARD))) (EQUAL X 'O) (BOARDP BOARD)) (EQUAL (MOVE X I BOARD) (NUMBER->BOARD (MOVEN (PIECE->NUMBER X) I (BOARD->NUMBER BOARD)) (LENGTH BOARD)))). This simplifies, expanding the functions NLISTP, ZEROP, and BOARDP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.8 0.1 ] MOVE-IS-MOVEN (DEFN NTHN (I BOARD) (IF (ZEROP I) (REMAINDER BOARD 4) (NTHN (SUB1 I) (QUOTIENT BOARD 4)))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform us that the measure (COUNT I) decreases according to the well-founded relation LESSP in each recursive call. Hence, NTHN is accepted under the principle of definition. Note that (NUMBERP (NTHN I BOARD)) is a theorem. [ 0.0 0.0 0.0 ] NTHN (PROVE-LEMMA NTH-IS-NTHN (REWRITE) (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (NTH I BOARD) (NUMBER->PIECE (NTHN I (BOARD->NUMBER BOARD)))))) This simplifies, expanding NUMBER->PIECE, to three new goals: Case 3. (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1)) (EQUAL (NTH I BOARD) 'X)), which again simplifies, opening up the definition of EQUAL, to the conjecture: (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1)) (EQUAL (NTH I BOARD) 'X)). Give the above formula the name *1. Case 2. (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1))) (EQUAL (NTH I BOARD) 'O)), which we would usually push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us consider: (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (NTH I BOARD) (NUMBER->PIECE (NTHN I (BOARD->NUMBER BOARD))))), which we named *1 above. We will appeal to induction. The recursive terms in the conjecture suggest six inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (p I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p (SUB1 I) (CDR BOARD))) (p I BOARD))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT can be used to establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme produces five new formulas: Case 5. (IMPLIES (AND (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (NTH I BOARD) (NUMBER->PIECE (NTHN I (BOARD->NUMBER BOARD))))), which simplifies, expanding the definitions of LENGTH, NOT, OR, EQUAL, and LESSP, to: T. Case 4. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL I 0) (NOT (NUMBERP I))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (NTH I BOARD) (NUMBER->PIECE (NTHN I (BOARD->NUMBER BOARD))))), which simplifies, rewriting with REMAINDER-PLUS-TIMES and REMAINDER-PLUS-TIMES-INSTANCE, and expanding the functions LENGTH, NOT, OR, EQUAL, LESSP, BOARDP, NTH, BOARD->NUMBER, PIECE->NUMBER, NTHN, NUMBER->PIECE, and PLUS, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (LESSP (SUB1 I) (LENGTH (CDR BOARD)))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (NTH I BOARD) (NUMBER->PIECE (NTHN I (BOARD->NUMBER BOARD))))). This simplifies, applying SUB1-ADD1, and unfolding the functions LENGTH, NOT, OR, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (BOARDP (CDR BOARD))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (NTH I BOARD) (NUMBER->PIECE (NTHN I (BOARD->NUMBER BOARD))))), which simplifies, applying the lemma SUB1-ADD1, and unfolding LENGTH, NOT, OR, LESSP, and BOARDP, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (EQUAL (NTH (SUB1 I) (CDR BOARD)) (NUMBER->PIECE (NTHN (SUB1 I) (BOARD->NUMBER (CDR BOARD))))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (NTH I BOARD) (NUMBER->PIECE (NTHN I (BOARD->NUMBER BOARD))))), which simplifies, applying SUB1-ADD1, QUOTIENT-PLUS-TIMES, and QUOTIENT-PLUS-TIMES-INSTANCE, and unfolding the functions LENGTH, NOT, OR, NUMBER->PIECE, LESSP, BOARDP, NTH, BOARD->NUMBER, PIECE->NUMBER, NTHN, EQUAL, and PLUS, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.1 ] NTH-IS-NTHN (DEFN LEGALPN (I BOARD) (AND (LESSP I 9) (EQUAL (NTHN I BOARD) 0))) Note that (OR (FALSEP (LEGALPN I BOARD)) (TRUEP (LEGALPN I BOARD))) is a theorem. [ 0.0 0.0 0.0 ] LEGALPN (PROVE-LEMMA LEGALP-IS-LEGALPN (REWRITE) (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (EQUAL (LEGALP I BOARD) (LEGALPN I (BOARD->NUMBER BOARD)))) ((ENABLE LEGALP))) WARNING: Note that the rewrite rule LEGALP-IS-LEGALPN will be stored so as to apply only to terms with the nonrecursive function symbol LEGALP. This simplifies, applying NTH-IS-NTHN, and unfolding NUMBER->PIECE, LEGALP, and LEGALPN, to the following two new formulas: Case 2. (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0))) (EQUAL F (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0))). This again simplifies, trivially, to: T. Case 1. (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (LESSP I 9) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (EQUAL T (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0))). This again simplifies, opening up the definition of EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALP-IS-LEGALPN (DEFN WINP1N (X I J K BOARD) (AND (EQUAL X (NTHN I BOARD)) (EQUAL X (NTHN J BOARD)) (EQUAL X (NTHN K BOARD)))) Note that: (OR (FALSEP (WINP1N X I J K BOARD)) (TRUEP (WINP1N X I J K BOARD))) is a theorem. [ 0.0 0.0 0.0 ] WINP1N (PROVE-LEMMA LESSP-NTHN-3-LEMMA NIL (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (PIECE->NUMBER (NTH I BOARD)) (NTHN I (BOARD->NUMBER BOARD)))) ((DISABLE NTH-IS-NTHN))) This simplifies, expanding PIECE->NUMBER, to three new goals: Case 3. (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD) (NOT (NTH I BOARD))) (EQUAL 0 (NTHN I (BOARD->NUMBER BOARD)))), which we will name *1. Case 2. (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD) (NTH I BOARD) (EQUAL (NTH I BOARD) 'X)) (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD)))). This again simplifies, trivially, to the new conjecture: (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD) (EQUAL (NTH I BOARD) 'X)) (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD)))), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (PIECE->NUMBER (NTH I BOARD)) (NTHN I (BOARD->NUMBER BOARD)))). We named this *1. We will try to prove it by induction. There are six plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (p I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p I BOARD)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p (SUB1 I) (CDR BOARD))) (p I BOARD))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT can be used to establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme leads to the following five new goals: Case 5. (IMPLIES (AND (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD)))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (PIECE->NUMBER (NTH I BOARD)) (NTHN I (BOARD->NUMBER BOARD)))). This simplifies, opening up the definitions of LENGTH, NOT, OR, EQUAL, and LESSP, to: T. Case 4. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (OR (EQUAL I 0) (NOT (NUMBERP I))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (PIECE->NUMBER (NTH I BOARD)) (NTHN I (BOARD->NUMBER BOARD)))). This simplifies, rewriting with REMAINDER-PLUS-TIMES and REMAINDER-PLUS-TIMES-INSTANCE, and unfolding the definitions of LENGTH, NOT, OR, EQUAL, LESSP, BOARDP, NTH, PIECE->NUMBER, BOARD->NUMBER, NTHN, and PLUS, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (LESSP (SUB1 I) (LENGTH (CDR BOARD)))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (PIECE->NUMBER (NTH I BOARD)) (NTHN I (BOARD->NUMBER BOARD)))), which simplifies, rewriting with SUB1-ADD1, and expanding LENGTH, NOT, OR, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (BOARDP (CDR BOARD))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (PIECE->NUMBER (NTH I BOARD)) (NTHN I (BOARD->NUMBER BOARD)))). This simplifies, applying SUB1-ADD1, and opening up the definitions of LENGTH, NOT, OR, LESSP, and BOARDP, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH BOARD) 0) (NOT (NUMBERP (LENGTH BOARD))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (EQUAL (PIECE->NUMBER (NTH (SUB1 I) (CDR BOARD))) (NTHN (SUB1 I) (BOARD->NUMBER (CDR BOARD)))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (EQUAL (PIECE->NUMBER (NTH I BOARD)) (NTHN I (BOARD->NUMBER BOARD)))), which simplifies, applying SUB1-ADD1, QUOTIENT-PLUS-TIMES, and QUOTIENT-PLUS-TIMES-INSTANCE, and expanding the definitions of LENGTH, NOT, OR, PIECE->NUMBER, LESSP, BOARDP, NTH, BOARD->NUMBER, NTHN, EQUAL, and PLUS, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.0 ] LESSP-NTHN-3-LEMMA (PROVE-LEMMA LESSP-NTHN-3 (REWRITE) (IMPLIES (AND (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (LESSP (NTHN I (BOARD->NUMBER BOARD)) 3)) ((USE (LESSP-NTHN-3-LEMMA)))) WARNING: Note that the proposed lemma LESSP-NTHN-3 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This conjecture simplifies, rewriting with NTH-IS-NTHN, and expanding the functions AND, NUMBER->PIECE, and IMPLIES, to three new goals: Case 3. (IMPLIES (AND (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1) (EQUAL (PIECE->NUMBER 'X) (NTHN I (BOARD->NUMBER BOARD))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (LESSP (PIECE->NUMBER 'X) 3)), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1)) (EQUAL (PIECE->NUMBER 'O) (NTHN I (BOARD->NUMBER BOARD))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (LESSP (PIECE->NUMBER 'O) 3)), which again simplifies, unfolding PIECE->NUMBER and LESSP, to: T. Case 1. (IMPLIES (AND (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0) (EQUAL (PIECE->NUMBER F) (NTHN I (BOARD->NUMBER BOARD))) (LESSP I (LENGTH BOARD)) (BOARDP BOARD)) (LESSP (PIECE->NUMBER F) 3)), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LESSP-NTHN-3 (PROVE-LEMMA WINP1-IS-WINP1N (REWRITE) (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (OR (EQUAL X 'X) (EQUAL X 'O))) (EQUAL (WINP1 X (LIST I J K) BOARD) (WINP1N (PIECE->NUMBER X) I J K (BOARD->NUMBER BOARD))))) This formula simplifies, rewriting with CDR-CONS, NTH-IS-NTHN, and CAR-CONS, and unfolding the definitions of OR, NUMBER->PIECE, WINP1, LISTP, PIECE->NUMBER, and WINP1N, to 18 new conjectures: Case 18.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (NOT (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD)))) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 1) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 0))) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 1))), which again simplifies, clearly, to: T. Case 17.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (NOT (EQUAL 1 (NTHN J (BOARD->NUMBER BOARD)))) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 1) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 0))) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 1))). This again simplifies, trivially, to: T. Case 16.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 1 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 0)) (EQUAL F (EQUAL 1 (NTHN K (BOARD->NUMBER BOARD))))). However this again simplifies, using linear arithmetic, to: T. Case 15.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 1 (NTHN J (BOARD->NUMBER BOARD))) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 1))) (EQUAL F (EQUAL 1 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, clearly, to: T. Case 14.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 1 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 0)) (EQUAL F (EQUAL 1 (NTHN K (BOARD->NUMBER BOARD))))). However this again simplifies, unfolding EQUAL, to: T. Case 13.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 1 (NTHN J (BOARD->NUMBER BOARD))) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 1))) (EQUAL F (EQUAL 1 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, obviously, to: T. Case 12.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 1 (NTHN J (BOARD->NUMBER BOARD))) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1))) (EQUAL F (EQUAL 1 (NTHN K (BOARD->NUMBER BOARD))))). This again simplifies, obviously, to: T. Case 11.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (EQUAL 1 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 1 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (EQUAL F (EQUAL 1 (NTHN K (BOARD->NUMBER BOARD))))). However this again simplifies, using linear arithmetic, to: T. Case 10.(IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'X) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 1) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 0)) (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 1)) (EQUAL T (EQUAL 1 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, opening up EQUAL, to: T. Case 9. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (NOT (EQUAL 2 (NTHN I (BOARD->NUMBER BOARD)))) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1)) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 1)) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 0))) (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 1)), which again simplifies, using linear arithmetic and rewriting with the lemma LESSP-NTHN-3, to: T. Case 8. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (NOT (EQUAL 2 (NTHN J (BOARD->NUMBER BOARD)))) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1)) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 1)) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 0))) (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 1)), which again simplifies, using linear arithmetic and appealing to the lemma LESSP-NTHN-3, to: T. Case 7. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (EQUAL 2 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 2 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1)) (EQUAL F (EQUAL 2 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, using linear arithmetic, to: T. Case 6. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (EQUAL 2 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 2 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 0)) (EQUAL F (EQUAL 2 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, using linear arithmetic, to: T. Case 5. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (EQUAL 2 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 2 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 0)) (EQUAL F (EQUAL 2 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, opening up EQUAL, to: T. Case 4. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (EQUAL 2 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 2 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 1)) (EQUAL F (EQUAL 2 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, expanding EQUAL, to: T. Case 3. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (EQUAL 2 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 2 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 1)) (EQUAL F (EQUAL 2 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (EQUAL 2 (NTHN I (BOARD->NUMBER BOARD))) (EQUAL 2 (NTHN J (BOARD->NUMBER BOARD))) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (EQUAL F (EQUAL 2 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (EQUAL X 'O) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 1)) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 1)) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 0)) (NOT (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 1))) (EQUAL T (EQUAL 2 (NTHN K (BOARD->NUMBER BOARD))))), which again simplifies, using linear arithmetic and applying LESSP-NTHN-3, to: (IMPLIES (AND (EQUAL (NTHN K (BOARD->NUMBER BOARD)) 2) (EQUAL (NTHN J (BOARD->NUMBER BOARD)) 2) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 2) (BOARDP BOARD) (LESSP I (LENGTH BOARD)) (LESSP J (LENGTH BOARD)) (LESSP K (LENGTH BOARD)) (NOT (EQUAL 2 0)) (NOT (EQUAL 2 1)) (NOT (EQUAL 2 0)) (NOT (EQUAL 2 1)) (NOT (EQUAL 2 0)) (NOT (EQUAL 2 1))) (EQUAL T (EQUAL 2 2))), which again simplifies, opening up EQUAL, to: T. Q.E.D. [ 0.0 0.3 0.0 ] WINP1-IS-WINP1N (DEFN WINPN (X BOARD) (OR (WINP1N X 0 1 2 BOARD) (WINP1N X 3 4 5 BOARD) (WINP1N X 6 7 8 BOARD) (WINP1N X 0 3 6 BOARD) (WINP1N X 1 4 7 BOARD) (WINP1N X 2 5 8 BOARD) (WINP1N X 0 4 8 BOARD) (WINP1N X 2 4 6 BOARD))) From the definition we can conclude that: (OR (FALSEP (WINPN X BOARD)) (TRUEP (WINPN X BOARD))) is a theorem. [ 0.0 0.0 0.0 ] WINPN (PROVE-LEMMA WINP-IS-WINPN (REWRITE) (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (OR (EQUAL X 'X) (EQUAL X 'O))) (EQUAL (WINP X BOARD) (WINPN (PIECE->NUMBER X) (BOARD->NUMBER BOARD)))) ((ENABLE WINP) (DISABLE PIECE->NUMBER *1*PIECE->NUMBER WINP1 WINP1N))) This conjecture simplifies, appealing to the lemma WINP1-IS-WINP1N, and opening up OR, LESSP, EQUAL, WINP, and WINPN, to: T. Q.E.D. [ 0.0 0.0 0.0 ] WINP-IS-WINPN (PROVE-LEMMA QUOTIENT-4-DECREASES (REWRITE) (IMPLIES (NOT (ZEROP N)) (LESSP (QUOTIENT N 4) N))) WARNING: Note that the proposed lemma QUOTIENT-4-DECREASES is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZEROP, NOT, and IMPLIES, to: (IMPLIES (AND (NOT (EQUAL N 0)) (NUMBERP N)) (LESSP (QUOTIENT N 4) N)). Applying the lemma REMAINDER-QUOTIENT-ELIM, replace N by (PLUS Z (TIMES 4 X)) to eliminate (QUOTIENT N 4) and (REMAINDER N 4). We use LESSP-REMAINDER, the type restriction lemma noted when QUOTIENT was introduced, and the type restriction lemma noted when REMAINDER was introduced to restrict the new variables. We would thus like to prove the new conjecture: (IMPLIES (AND (NUMBERP X) (NUMBERP Z) (EQUAL (LESSP Z 4) (NOT (ZEROP 4))) (NOT (EQUAL (PLUS Z (TIMES 4 X)) 0))) (LESSP X (PLUS Z (TIMES 4 X)))), which simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] QUOTIENT-4-DECREASES (DEFN PICK-MOVE1N (BOARD I LEN) (COND ((ZEROP LEN) 0) ((ZEROP (REMAINDER BOARD 4)) I) (T (PICK-MOVE1N (QUOTIENT BOARD 4) (ADD1 I) (SUB1 LEN))))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to show that the measure (COUNT LEN) decreases according to the well-founded relation LESSP in each recursive call. Hence, PICK-MOVE1N is accepted under the definitional principle. Note that: (OR (NUMBERP (PICK-MOVE1N BOARD I LEN)) (EQUAL (PICK-MOVE1N BOARD I LEN) I)) is a theorem. [ 0.0 0.0 0.0 ] PICK-MOVE1N (DEFN BOARD->NUMBER-ALIST (ALIST) (IF (NLISTP ALIST) NIL (CONS (CONS (BOARD->NUMBER (CAAR ALIST)) (CDAR ALIST)) (BOARD->NUMBER-ALIST (CDR ALIST))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT ALIST) decreases according to the well-founded relation LESSP in each recursive call. Hence, BOARD->NUMBER-ALIST is accepted under the principle of definition. Note that: (OR (LITATOM (BOARD->NUMBER-ALIST ALIST)) (LISTP (BOARD->NUMBER-ALIST ALIST))) is a theorem. [ 0.0 0.0 0.1 ] BOARD->NUMBER-ALIST (DEFN BOARDP-ALISTP (ALIST) (IF (NLISTP ALIST) (EQUAL ALIST NIL) (AND (BOARDP (CAAR ALIST)) (EQUAL (LENGTH (CAAR ALIST)) 9) (BOARDP-ALISTP (CDR ALIST))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT ALIST) decreases according to the well-founded relation LESSP in each recursive call. Hence, BOARDP-ALISTP is accepted under the principle of definition. Note that: (OR (FALSEP (BOARDP-ALISTP ALIST)) (TRUEP (BOARDP-ALISTP ALIST))) is a theorem. [ 0.0 0.0 0.0 ] BOARDP-ALISTP (DEFN PICK-MOVEN (BOARD) (LET ((BOOK-MOVE (ASSOC BOARD '((91648 . 2) (90688 . 2) (92692 . 3) (92689 . 3) (90640 . 5) (90628 . 2) (90625 . 2) (82432 . 6) (103936 . 1) (102976 . 1) (102928 . 1) (102934 . 5) (70144 . 7) (83488 . 6) (104032 . 1) (71200 . 7) (67168 . 6) (67108 . 6) (67105 . 6) (67072 . 2) (90688 . 2) (75328 . 2) (74320 . 5) (74308 . 2) (74305 . 2) (66112 . 6) (84496 . 3) (72208 . 3) (84562 . 6) (72274 . 7) (68116 . 3) (68113 . 3) (66064 . 5) (82468 . 6) (70180 . 7) (67108 . 6) (66148 . 6) (66085 . 6) (66052 . 2) (90649 . 5) (82441 . 6) (70153 . 7) (67081 . 7) (66121 . 7) (66073 . 7) (65536 . 4) (91648 . 2) (90688 . 2) (92692 . 3) (92689 . 3) (90640 . 5) (90628 . 2) (90625 . 2) (82432 . 6) (152209 . 5) (152197 . 5) (152065 . 3) (20992 . 8) (153097 . 3) (149065 . 6) (17920 . 8) (90688 . 2) (26176 . 2) (90706 . 5) (26194 . 8) (25174 . 8) (25156 . 2) (25153 . 2) (16960 . 6) (148057 . 6) (16912 . 8) (82468 . 6) (152164 . 5) (152101 . 5) (21028 . 8) (17956 . 6) (16996 . 6) (16933 . 6) (16900 . 2) (90625 . 2) (26113 . 2) (25153 . 2) (90649 . 5) (26137 . 8) (16897 . 6) (16384 . 4) (103936 . 1) (102976 . 1) (102928 . 1) (102934 . 5) (70144 . 7) (152209 . 5) (152197 . 5) (152065 . 3) (20992 . 8) (136705 . 3) (5632 . 8) (70210 . 7) (21058 . 8) (5698 . 8) (4690 . 8) (4678 . 8) (70168 . 7) (21016 . 8) (5656 . 7) (4696 . 7) (4633 . 7) (4624 . 1) (102934 . 5) (70150 . 7) (20998 . 8) (5638 . 8) (4678 . 8) (4630 . 8) (70273 . 5) (21121 . 5) (71305 . 7) (22153 . 8) (4753 . 5) (4741 . 5) (4609 . 3) (4096 . 4) (83488 . 6) (104032 . 1) (71200 . 7) (67168 . 6) (67108 . 6) (67105 . 6) (67072 . 2) (153097 . 3) (149065 . 6) (17920 . 8) (136705 . 3) (5632 . 8) (67138 . 2) (17986 . 8) (5698 . 8) (1618 . 8) (1606 . 8) (136729 . 7) (132697 . 7) (1552 . 8) (67108 . 6) (17956 . 6) (71206 . 7) (22054 . 8) (5734 . 8) (1636 . 6) (1573 . 6) (1540 . 2) (67105 . 6) (17953 . 6) (71329 . 7) (22177 . 8) (5665 . 3) (1633 . 6) (1573 . 6) (1537 . 2) (1024 . 4) (65830 . 7) (65794 . 2) (16666 . 6) (4390 . 7) (4354 . 2) (66946 . 6) (17794 . 6) (5506 . 2) (1426 . 6) (1414 . 6) (1282 . 3) (6502 . 8) (6466 . 2) (2386 . 6) (2374 . 7) (322 . 5) (74002 . 3) (24850 . 3) (9490 . 3) (8530 . 5) (274 . 6) (99718 . 6) (34054 . 3) (33094 . 5) (42262 . 8) (41302 . 8) (33046 . 6) (262 . 7) (90688 . 2) (75328 . 2) (74320 . 5) (74308 . 2) (74305 . 2) (66112 . 6) (90688 . 2) (26176 . 2) (90706 . 5) (26194 . 8) (25174 . 8) (25156 . 2) (25153 . 2) (16960 . 6) (70210 . 7) (21058 . 8) (5698 . 8) (4690 . 8) (4678 . 8) (67138 . 2) (17986 . 8) (5698 . 8) (1618 . 8) (1606 . 8) (84562 . 6) (72274 . 7) (66130 . 5) (16978 . 8) (4690 . 8) (1618 . 8) (598 . 8) (82534 . 6) (70246 . 7) (16966 . 8) (4678 . 8) (1606 . 8) (598 . 8) (74305 . 2) (25153 . 2) (9793 . 2) (74329 . 7) (577 . 6) (64 . 4) (84496 . 3) (72208 . 3) (84562 . 6) (72274 . 7) (68116 . 3) (68113 . 3) (66064 . 5) (148057 . 6) (16912 . 8) (70168 . 7) (21016 . 8) (5656 . 7) (4696 . 7) (4633 . 7) (4624 . 1) (136729 . 7) (132697 . 7) (1552 . 8) (84562 . 6) (72274 . 7) (66130 . 5) (16978 . 8) (4690 . 8) (1618 . 8) (598 . 8) (66070 . 5) (16918 . 8) (4630 . 8) (1558 . 8) (598 . 8) (66073 . 7) (18073 . 8) (4633 . 7) (1561 . 7) (601 . 7) (16 . 4) (82468 . 6) (70180 . 7) (67108 . 6) (66148 . 6) (66085 . 6) (66052 . 2) (82468 . 6) (152164 . 5) (152101 . 5) (21028 . 8) (17956 . 6) (16996 . 6) (16933 . 6) (16900 . 2) (102934 . 5) (70150 . 7) (20998 . 8) (5638 . 8) (4678 . 8) (4630 . 8) (67108 . 6) (17956 . 6) (71206 . 7) (22054 . 8) (5734 . 8) (1636 . 6) (1573 . 6) (1540 . 2) (82534 . 6) (70246 . 7) (16966 . 8) (4678 . 8) (1606 . 8) (598 . 8) (66070 . 5) (16918 . 8) (4630 . 8) (1558 . 8) (598 . 8) (66085 . 6) (16933 . 6) (1573 . 6) (613 . 6) (4 . 4) (90649 . 5) (82441 . 6) (70153 . 7) (67081 . 7) (66121 . 7) (66073 . 7) (90625 . 2) (26113 . 2) (25153 . 2) (90649 . 5) (26137 . 8) (16897 . 6) (70273 . 5) (21121 . 5) (71305 . 7) (22153 . 8) (4753 . 5) (4741 . 5) (4609 . 3) (67105 . 6) (17953 . 6) (71329 . 7) (22177 . 8) (5665 . 3) (1633 . 6) (1573 . 6) (1537 . 2) (74305 . 2) (25153 . 2) (9793 . 2) (74329 . 7) (577 . 6) (66073 . 7) (18073 . 8) (4633 . 7) (1561 . 7) (601 . 7) (66085 . 6) (16933 . 6) (1573 . 6) (613 . 6) (1 . 4))))) (IF BOOK-MOVE (CDR BOOK-MOVE) (PICK-MOVE1N BOARD 0 9)))) [ 0.0 0.0 0.0 ] PICK-MOVEN (COMPILE-UNCOMPILED-DEFNS "tmp") Loading ./basic/tmp.o Finished loading ./basic/tmp.o /v/hank/v28/boyer/nqthm-2nd/nqthm-1992/examples/basic/tmp.lisp (PROVE-LEMMA BOARD->NUMBER-ONE-TO-ONE (REWRITE) (IMPLIES (AND (EQUAL (LENGTH BOARD1) (LENGTH BOARD2)) (BOARDP BOARD1) (BOARDP BOARD2)) (EQUAL (EQUAL (BOARD->NUMBER BOARD1) (BOARD->NUMBER BOARD2)) (EQUAL BOARD1 BOARD2))) ((DISABLE BOARD->NUMBER->BOARD) (USE (BOARD->NUMBER->BOARD (BOARD BOARD1)) (BOARD->NUMBER->BOARD (BOARD BOARD2))))) This conjecture simplifies, expanding the definition of IMPLIES, to two new goals: Case 2. (IMPLIES (AND (EQUAL (NUMBER->BOARD (BOARD->NUMBER BOARD1) (LENGTH BOARD1)) BOARD1) (EQUAL (NUMBER->BOARD (BOARD->NUMBER BOARD2) (LENGTH BOARD1)) BOARD2) (EQUAL (LENGTH BOARD1) (LENGTH BOARD2)) (BOARDP BOARD1) (BOARDP BOARD2) (NOT (EQUAL BOARD1 BOARD2))) (NOT (EQUAL (BOARD->NUMBER BOARD1) (BOARD->NUMBER BOARD2)))), which again simplifies, clearly, to: T. Case 1. (IMPLIES (AND (EQUAL (NUMBER->BOARD (BOARD->NUMBER BOARD1) (LENGTH BOARD1)) BOARD1) (EQUAL (NUMBER->BOARD (BOARD->NUMBER BOARD2) (LENGTH BOARD1)) BOARD2) (EQUAL (LENGTH BOARD1) (LENGTH BOARD2)) (BOARDP BOARD1) (BOARDP BOARD2) (EQUAL BOARD1 BOARD2)) (EQUAL (EQUAL (BOARD->NUMBER BOARD1) (BOARD->NUMBER BOARD2)) T)). However this again simplifies, opening up the function EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] BOARD->NUMBER-ONE-TO-ONE (PROVE-LEMMA ASSOC-BOARD->NUMBER (REWRITE) (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (IFF (ASSOC BOARD ALIST) (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST))))) This conjecture simplifies, expanding IFF, to the following two new conjectures: Case 2. (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST) (NOT (ASSOC BOARD ALIST))) (NOT (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST)))). Name the above subgoal *1. Case 1. (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST) (ASSOC BOARD ALIST)) (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST))), which we would usually push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us consider: (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (IFF (ASSOC BOARD ALIST) (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST)))). We gave this the name *1 above. Perhaps we can prove it by induction. There are six plausible inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (NLISTP ALIST) (p BOARD ALIST)) (IMPLIES (AND (NOT (NLISTP ALIST)) (EQUAL BOARD (CAAR ALIST))) (p BOARD ALIST)) (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (p BOARD (CDR ALIST))) (p BOARD ALIST))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to show that the measure (COUNT ALIST) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates four new goals: Case 4. (IMPLIES (AND (NLISTP ALIST) (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (IFF (ASSOC BOARD ALIST) (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST)))), which simplifies, unfolding the functions NLISTP, BOARDP-ALISTP, LISTP, ASSOC, BOARD->NUMBER-ALIST, and IFF, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP ALIST)) (EQUAL BOARD (CAAR ALIST)) (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (IFF (ASSOC BOARD ALIST) (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST)))), which simplifies, applying CAR-CONS, and opening up the functions NLISTP, BOARDP-ALISTP, EQUAL, ASSOC, BOARD->NUMBER-ALIST, and IFF, to: (IMPLIES (AND (LISTP ALIST) (EQUAL (LENGTH (CAAR ALIST)) 9) (BOARDP (CAAR ALIST)) (BOARDP-ALISTP (CDR ALIST))) (CAR ALIST)), which again simplifies, expanding the functions CAR, LENGTH, and EQUAL, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (NOT (BOARDP-ALISTP (CDR ALIST))) (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (IFF (ASSOC BOARD ALIST) (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST)))), which simplifies, expanding NLISTP and BOARDP-ALISTP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (IFF (ASSOC BOARD (CDR ALIST)) (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST (CDR ALIST)))) (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (IFF (ASSOC BOARD ALIST) (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST)))), which simplifies, applying CDR-CONS, BOARD->NUMBER-ONE-TO-ONE, and CAR-CONS, and unfolding the definitions of NLISTP, IFF, BOARDP-ALISTP, ASSOC, BOARD->NUMBER-ALIST, and EQUAL, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] ASSOC-BOARD->NUMBER (PROVE-LEMMA CDR-ASSOC-BOARD->NUMBER (REWRITE) (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (EQUAL (CDR (ASSOC BOARD ALIST)) (CDR (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST)))))) Give the conjecture the name *1. We will appeal to induction. There are six plausible inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (NLISTP ALIST) (p BOARD ALIST)) (IMPLIES (AND (NOT (NLISTP ALIST)) (EQUAL BOARD (CAAR ALIST))) (p BOARD ALIST)) (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (p BOARD (CDR ALIST))) (p BOARD ALIST))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT ALIST) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates the following four new formulas: Case 4. (IMPLIES (AND (NLISTP ALIST) (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (EQUAL (CDR (ASSOC BOARD ALIST)) (CDR (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST))))). This simplifies, unfolding NLISTP, BOARDP-ALISTP, LISTP, ASSOC, CDR, BOARD->NUMBER-ALIST, and EQUAL, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP ALIST)) (EQUAL BOARD (CAAR ALIST)) (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (EQUAL (CDR (ASSOC BOARD ALIST)) (CDR (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST))))). This simplifies, applying the lemmas CAR-CONS and CDR-CONS, and unfolding the definitions of NLISTP, BOARDP-ALISTP, EQUAL, ASSOC, and BOARD->NUMBER-ALIST, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (NOT (BOARDP-ALISTP (CDR ALIST))) (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (EQUAL (CDR (ASSOC BOARD ALIST)) (CDR (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST))))). This simplifies, unfolding the definitions of NLISTP and BOARDP-ALISTP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP ALIST)) (NOT (EQUAL BOARD (CAAR ALIST))) (EQUAL (CDR (ASSOC BOARD (CDR ALIST))) (CDR (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST (CDR ALIST))))) (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD) (BOARDP-ALISTP ALIST)) (EQUAL (CDR (ASSOC BOARD ALIST)) (CDR (ASSOC (BOARD->NUMBER BOARD) (BOARD->NUMBER-ALIST ALIST))))). This simplifies, rewriting with the lemmas CDR-CONS, BOARD->NUMBER-ONE-TO-ONE, and CAR-CONS, and expanding NLISTP, BOARDP-ALISTP, ASSOC, BOARD->NUMBER-ALIST, and EQUAL, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.1 ] CDR-ASSOC-BOARD->NUMBER (PROVE-LEMMA PICK-MOVE1-IS-PICK-MOVE1N (REWRITE) (IMPLIES (BOARDP BOARD) (EQUAL (PICK-MOVE1 BOARD I) (PICK-MOVE1N (BOARD->NUMBER BOARD) I (LENGTH BOARD))))) Name the conjecture *1. Perhaps we can prove it by induction. Four inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (EQUAL (CAR BOARD) F)) (p BOARD I)) (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (EQUAL (CAR BOARD) F)) (p (CDR BOARD) (ADD1 I))) (p BOARD I))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for I. The above induction scheme leads to the following four new conjectures: Case 4. (IMPLIES (AND (NLISTP BOARD) (BOARDP BOARD)) (EQUAL (PICK-MOVE1 BOARD I) (PICK-MOVE1N (BOARD->NUMBER BOARD) I (LENGTH BOARD)))). This simplifies, rewriting with the lemma ENDP-IMPLIES-PICK-MOVE1-0, and opening up the definitions of NLISTP, BOARDP, ENDP, BOARD->NUMBER, LENGTH, EQUAL, and PICK-MOVE1N, to: T. Case 3. (IMPLIES (AND (NOT (NLISTP BOARD)) (EQUAL (CAR BOARD) F) (BOARDP BOARD)) (EQUAL (PICK-MOVE1 BOARD I) (PICK-MOVE1N (BOARD->NUMBER BOARD) I (LENGTH BOARD)))). This simplifies, appealing to the lemma REMAINDER-PLUS-TIMES-INSTANCE, and expanding NLISTP, BOARDP, EQUAL, PICK-MOVE1, BOARD->NUMBER, PIECE->NUMBER, PLUS, LENGTH, and PICK-MOVE1N, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (EQUAL (CAR BOARD) F)) (NOT (BOARDP (CDR BOARD))) (BOARDP BOARD)) (EQUAL (PICK-MOVE1 BOARD I) (PICK-MOVE1N (BOARD->NUMBER BOARD) I (LENGTH BOARD)))). This simplifies, expanding the functions NLISTP and BOARDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (EQUAL (CAR BOARD) F)) (EQUAL (PICK-MOVE1 (CDR BOARD) (ADD1 I)) (PICK-MOVE1N (BOARD->NUMBER (CDR BOARD)) (ADD1 I) (LENGTH (CDR BOARD)))) (BOARDP BOARD)) (EQUAL (PICK-MOVE1 BOARD I) (PICK-MOVE1N (BOARD->NUMBER BOARD) I (LENGTH BOARD)))). This simplifies, rewriting with SUB1-ADD1, QUOTIENT-PLUS-TIMES, and REMAINDER-PLUS-TIMES, and unfolding the definitions of NLISTP, BOARDP, PICK-MOVE1, EQUAL, BOARD->NUMBER, PIECE->NUMBER, LENGTH, LESSP, and PICK-MOVE1N, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] PICK-MOVE1-IS-PICK-MOVE1N (PROVE-LEMMA PICK-MOVE-IS-PICK-MOVEN (REWRITE) (IMPLIES (AND (EQUAL (LENGTH BOARD) 9) (BOARDP BOARD)) (EQUAL (PICK-MOVE BOARD) (PICK-MOVEN (BOARD->NUMBER BOARD)))) ((DISABLE ASSOC) (ENABLE PICK-MOVE))) This simplifies, appealing to the lemmas PICK-MOVE1-IS-PICK-MOVE1N, CDR-ASSOC-BOARD->NUMBER, and ASSOC-BOARD->NUMBER, and opening up the functions BOARD->NUMBER-ALIST, BOARDP-ALISTP, PICK-MOVE, and PICK-MOVEN, to: T. Q.E.D. [ 0.0 0.0 0.0 ] PICK-MOVE-IS-PICK-MOVEN (DISABLE PICK-MOVEN) [ 0.0 0.0 0.0 ] PICK-MOVEN-OFF (DEFN TIC-TAC-TOE1-B (XMOVES BOARD I) (LET ((BOARD1 (MOVEN 1 (CAR XMOVES) BOARD))) (IF (LEGALPN (CAR XMOVES) BOARD) (IF (LESSP I 9) (LET ((BOARD2 (MOVEN 2 (PICK-MOVEN BOARD1) BOARD1))) (IF (WINPN 2 BOARD2) (LIST BOARD1 BOARD2 -2) (CONS BOARD1 (CONS BOARD2 (TIC-TAC-TOE1-B (CDR XMOVES) BOARD2 (ADD1 (ADD1 I))))))) (LIST BOARD1 -1)) (LIST BOARD1 -2))) ((LESSP (DIFFERENCE 9 I)))) Linear arithmetic, the lemma SUB1-TYPE-RESTRICTION, and the definitions of LESSP, ADD1, DIFFERENCE, WINPN, NTHN, EQUAL, WINP1N, and LEGALPN establish that the measure (DIFFERENCE 9 I) decreases according to the well-founded relation LESSP in each recursive call. Hence, TIC-TAC-TOE1-B is accepted under the principle of definition. Note that: (LISTP (TIC-TAC-TOE1-B XMOVES BOARD I)) is a theorem. [ 0.0 0.1 0.0 ] TIC-TAC-TOE1-B (DEFN TIC-TAC-TOE-B (XMOVES) (CONS 0 (TIC-TAC-TOE1-B XMOVES 0 1))) Observe that (LISTP (TIC-TAC-TOE-B XMOVES)) is a theorem. [ 0.0 0.0 0.0 ] TIC-TAC-TOE-B (PROVE-LEMMA LEGALPN-IMPLIES-LESSP (REWRITE) (IMPLIES (AND (LEGALPN I (BOARD->NUMBER BOARD)) (EQUAL (LENGTH BOARD) 9)) (EQUAL (LESSP I 9) T))) WARNING: Note that LEGALPN-IMPLIES-LESSP contains the free variable BOARD which will be chosen by instantiating the hypothesis: (LEGALPN I (BOARD->NUMBER BOARD)). This conjecture can be simplified, using the abbreviations LEGALPN, AND, and IMPLIES, to: (IMPLIES (AND (LESSP I 9) (EQUAL (NTHN I (BOARD->NUMBER BOARD)) 0) (EQUAL (LENGTH BOARD) 9)) (EQUAL (LESSP I 9) T)). This simplifies, obviously, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LEGALPN-IMPLIES-LESSP (PROVE-LEMMA LENGTH-NUMBER->BOARD (REWRITE) (EQUAL (LENGTH (NUMBER->BOARD N LEN)) (FIX LEN))) This simplifies, unfolding FIX, to two new formulas: Case 2. (IMPLIES (NOT (NUMBERP LEN)) (EQUAL (LENGTH (NUMBER->BOARD N LEN)) 0)), which again simplifies, unfolding NUMBER->BOARD, LENGTH, and EQUAL, to: T. Case 1. (IMPLIES (NUMBERP LEN) (EQUAL (LENGTH (NUMBER->BOARD N LEN)) LEN)), which we will name *1. We will appeal to induction. There is only one plausible induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP LEN) (p N LEN)) (IMPLIES (AND (NOT (ZEROP LEN)) (p (QUOTIENT N 4) (SUB1 LEN))) (p N LEN))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform us that the measure (COUNT LEN) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for N. The above induction scheme generates the following two new goals: Case 2. (IMPLIES (AND (ZEROP LEN) (NUMBERP LEN)) (EQUAL (LENGTH (NUMBER->BOARD N LEN)) LEN)). This simplifies, opening up the definitions of ZEROP, NUMBERP, EQUAL, NUMBER->BOARD, and LENGTH, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP LEN)) (EQUAL (LENGTH (NUMBER->BOARD (QUOTIENT N 4) (SUB1 LEN))) (SUB1 LEN)) (NUMBERP LEN)) (EQUAL (LENGTH (NUMBER->BOARD N LEN)) LEN)). This simplifies, applying ADD1-SUB1 and CDR-CONS, and expanding ZEROP, NUMBER->BOARD, NUMBER->PIECE, and LENGTH, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LENGTH-NUMBER->BOARD (PROVE-LEMMA BOARDP-NUMBER-BOARD (REWRITE) (BOARDP (NUMBER->BOARD N LEN))) Name the conjecture *1. We will appeal to induction. There is only one plausible induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP LEN) (p N LEN)) (IMPLIES (AND (NOT (ZEROP LEN)) (p (QUOTIENT N 4) (SUB1 LEN))) (p N LEN))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to show that the measure (COUNT LEN) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for N. The above induction scheme leads to two new goals: Case 2. (IMPLIES (ZEROP LEN) (BOARDP (NUMBER->BOARD N LEN))), which simplifies, unfolding the definitions of ZEROP, EQUAL, NUMBER->BOARD, and BOARDP, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP LEN)) (BOARDP (NUMBER->BOARD (QUOTIENT N 4) (SUB1 LEN)))) (BOARDP (NUMBER->BOARD N LEN))), which simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and expanding ZEROP, NUMBER->BOARD, NUMBER->PIECE, and BOARDP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] BOARDP-NUMBER-BOARD (DEFN BOARDPN (N) (IF (ZEROP N) T (IF (LESSP (REMAINDER N 4) 3) (BOARDPN (QUOTIENT N 4)) F))) Linear arithmetic, the lemmas COUNT-NUMBERP and QUOTIENT-4-DECREASES, and the definition of ZEROP inform us that the measure (COUNT N) decreases according to the well-founded relation LESSP in each recursive call. Hence, BOARDPN is accepted under the definitional principle. Note that: (OR (FALSEP (BOARDPN N)) (TRUEP (BOARDPN N))) is a theorem. [ 0.0 0.0 0.0 ] BOARDPN (PROVE-LEMMA BOARDPN-BOARD->NUMBER (REWRITE) (IMPLIES (BOARDP BOARD) (BOARDPN (BOARD->NUMBER BOARD)))) Give the conjecture the name *1. We will appeal to induction. Two inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (p (CDR BOARD))) (p BOARD))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces the following three new conjectures: Case 3. (IMPLIES (AND (NLISTP BOARD) (BOARDP BOARD)) (BOARDPN (BOARD->NUMBER BOARD))). This simplifies, expanding the functions NLISTP, BOARDP, BOARD->NUMBER, and BOARDPN, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (BOARDP (CDR BOARD))) (BOARDP BOARD)) (BOARDPN (BOARD->NUMBER BOARD))). This simplifies, opening up NLISTP and BOARDP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (BOARDPN (BOARD->NUMBER (CDR BOARD))) (BOARDP BOARD)) (BOARDPN (BOARD->NUMBER BOARD))). This simplifies, rewriting with QUOTIENT-PLUS-TIMES, REMAINDER-PLUS-TIMES, QUOTIENT-PLUS-TIMES-INSTANCE, and REMAINDER-PLUS-TIMES-INSTANCE, and expanding NLISTP, BOARDP, BOARD->NUMBER, PIECE->NUMBER, LESSP, BOARDPN, EQUAL, and PLUS, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] BOARDPN-BOARD->NUMBER (PROVE-LEMMA BOARDPN-MOVEN (REWRITE) (IMPLIES (AND (NUMBERP BOARD) (BOARDPN BOARD) (OR (EQUAL X 1) (EQUAL X 2))) (BOARDPN (MOVEN X I BOARD))) ((DISABLE TIMES))) This simplifies, expanding the definition of OR, to two new goals: Case 2. (IMPLIES (AND (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 1)) (BOARDPN (MOVEN 1 I BOARD))), which again simplifies, obviously, to: (IMPLIES (AND (NUMBERP BOARD) (BOARDPN BOARD)) (BOARDPN (MOVEN 1 I BOARD))), which we will name *1. Case 1. (IMPLIES (AND (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 2)) (BOARDPN (MOVEN 2 I BOARD))). This again simplifies, clearly, to the new formula: (IMPLIES (AND (NUMBERP BOARD) (BOARDPN BOARD)) (BOARDPN (MOVEN 2 I BOARD))), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (AND (IMPLIES (AND (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 2)) (BOARDPN (MOVEN X I BOARD))) (IMPLIES (AND (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 1)) (BOARDPN (MOVEN X I BOARD)))), named *1. Let us appeal to the induction principle. Four inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP BOARD) (p X I BOARD)) (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (p X (SUB1 I) (QUOTIENT BOARD 4))) (p X I BOARD)) (IMPLIES (AND (NOT (ZEROP BOARD)) (NOT (LESSP (REMAINDER BOARD 4) 3))) (p X I BOARD))). Linear arithmetic, the lemmas COUNT-NUMBERP and QUOTIENT-4-DECREASES, and the definition of ZEROP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for I. The above induction scheme generates eight new goals: Case 8. (IMPLIES (AND (ZEROP BOARD) (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 2)) (BOARDPN (MOVEN X I BOARD))), which simplifies, opening up the functions ZEROP, NUMBERP, and BOARDPN, to the formula: (IMPLIES (EQUAL BOARD 0) (BOARDPN (MOVEN 2 I 0))). This again simplifies, obviously, to: (BOARDPN (MOVEN 2 I 0)), which we will name *1.1. Case 7. (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (NOT (BOARDPN (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 2)) (BOARDPN (MOVEN X I BOARD))). This simplifies, expanding the functions ZEROP and BOARDPN, to: T. Case 6. (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (BOARDPN (MOVEN X (SUB1 I) (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 2)) (BOARDPN (MOVEN X I BOARD))). This simplifies, opening up the functions ZEROP, BOARDPN, and MOVEN, to the following three new formulas: Case 6.3. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (BOARDPN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL I 0)) (NUMBERP I)) (BOARDPN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4)))))). This again simplifies, using linear arithmetic, rewriting with the lemmas QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and opening up the function BOARDPN, to: T. Case 6.2. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (BOARDPN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (EQUAL I 0)) (BOARDPN (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))))), which again simplifies, rewriting with QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and expanding the definitions of SUB1, EQUAL, MOVEN, LESSP, and BOARDPN, to: T. Case 6.1. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (BOARDPN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (NUMBERP I))) (BOARDPN (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))))). But this again simplifies, applying SUB1-NNUMBERP, QUOTIENT-PLUS-TIMES, and REMAINDER-PLUS-TIMES, and expanding the functions EQUAL, MOVEN, LESSP, and BOARDPN, to: T. Case 5. (IMPLIES (AND (NOT (ZEROP BOARD)) (NOT (LESSP (REMAINDER BOARD 4) 3)) (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 2)) (BOARDPN (MOVEN X I BOARD))). This simplifies, expanding the definitions of ZEROP and BOARDPN, to: T. Case 4. (IMPLIES (AND (ZEROP BOARD) (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 1)) (BOARDPN (MOVEN X I BOARD))). This simplifies, unfolding the definitions of ZEROP, NUMBERP, and BOARDPN, to the new goal: (IMPLIES (EQUAL BOARD 0) (BOARDPN (MOVEN 1 I 0))), which again simplifies, obviously, to the new conjecture: (BOARDPN (MOVEN 1 I 0)), which we will name *1.2. Case 3. (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (NOT (BOARDPN (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 1)) (BOARDPN (MOVEN X I BOARD))). This simplifies, unfolding the functions ZEROP and BOARDPN, to: T. Case 2. (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (BOARDPN (MOVEN X (SUB1 I) (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 1)) (BOARDPN (MOVEN X I BOARD))). This simplifies, unfolding ZEROP, BOARDPN, and MOVEN, to the following three new conjectures: Case 2.3. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (BOARDPN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL I 0)) (NUMBERP I)) (BOARDPN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4)))))). However this again simplifies, using linear arithmetic, applying QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and opening up the function BOARDPN, to: T. Case 2.2. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (BOARDPN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (EQUAL I 0)) (BOARDPN (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))))). However this again simplifies, appealing to the lemmas QUOTIENT-PLUS-TIMES and REMAINDER-PLUS-TIMES, and opening up SUB1, EQUAL, MOVEN, LESSP, and BOARDPN, to: T. Case 2.1. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (BOARDPN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (NUMBERP I))) (BOARDPN (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))))), which again simplifies, applying SUB1-NNUMBERP, QUOTIENT-PLUS-TIMES, and REMAINDER-PLUS-TIMES, and opening up EQUAL, MOVEN, LESSP, and BOARDPN, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP BOARD)) (NOT (LESSP (REMAINDER BOARD 4) 3)) (NUMBERP BOARD) (BOARDPN BOARD) (EQUAL X 1)) (BOARDPN (MOVEN X I BOARD))). This simplifies, unfolding the functions ZEROP and BOARDPN, to: T. So next consider: (BOARDPN (MOVEN 1 I 0)), named *1.2 above. We will appeal to induction. There is only one plausible induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP I) (p I)) (IMPLIES (AND (NOT (ZEROP I)) (p (SUB1 I))) (p I))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to the following two new goals: Case 2. (IMPLIES (ZEROP I) (BOARDPN (MOVEN 1 I 0))). This simplifies, unfolding the functions ZEROP, MOVEN, BOARDPN, QUOTIENT, TIMES, and PLUS, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP I)) (BOARDPN (MOVEN 1 (SUB1 I) 0))) (BOARDPN (MOVEN 1 I 0))). This simplifies, applying QUOTIENT-PLUS-TIMES-INSTANCE and REMAINDER-PLUS-TIMES-INSTANCE, and unfolding the functions ZEROP, MOVEN, REMAINDER, QUOTIENT, EQUAL, PLUS, LESSP, and BOARDPN, to: T. That finishes the proof of *1.2. So let us turn our attention to: (BOARDPN (MOVEN 2 I 0)), which is formula *1.1 above. Let us appeal to the induction principle. There is only one suggested induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP I) (p I)) (IMPLIES (AND (NOT (ZEROP I)) (p (SUB1 I))) (p I))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme leads to the following two new goals: Case 2. (IMPLIES (ZEROP I) (BOARDPN (MOVEN 2 I 0))). This simplifies, opening up ZEROP, MOVEN, BOARDPN, QUOTIENT, TIMES, and PLUS, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP I)) (BOARDPN (MOVEN 2 (SUB1 I) 0))) (BOARDPN (MOVEN 2 I 0))). This simplifies, rewriting with the lemmas QUOTIENT-PLUS-TIMES-INSTANCE and REMAINDER-PLUS-TIMES-INSTANCE, and expanding the definitions of ZEROP, MOVEN, REMAINDER, QUOTIENT, EQUAL, PLUS, LESSP, and BOARDPN, to: T. That finishes the proof of *1.1, which, consequently, also finishes the proof of *1. Q.E.D. [ 0.0 0.3 0.0 ] BOARDPN-MOVEN (DEFN LENGTHN (BOARD) (IF (ZEROP BOARD) 0 (ADD1 (LENGTHN (QUOTIENT BOARD 4))))) Linear arithmetic, the lemmas COUNT-NUMBERP and QUOTIENT-4-DECREASES, and the definition of ZEROP inform us that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each recursive call. Hence, LENGTHN is accepted under the definitional principle. From the definition we can conclude that (NUMBERP (LENGTHN BOARD)) is a theorem. [ 0.0 0.0 0.0 ] LENGTHN (PROVE-LEMMA NUMBER->BOARD->NUMBER (REWRITE) (IMPLIES (AND (NUMBERP BOARD) (BOARDPN BOARD) (NOT (LESSP LEN (LENGTHN BOARD)))) (EQUAL (BOARD->NUMBER (NUMBER->BOARD BOARD LEN)) BOARD))) Give the conjecture the name *1. Let us appeal to the induction principle. There are four plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP BOARD) (p BOARD LEN)) (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (p (QUOTIENT BOARD 4) (SUB1 LEN))) (p BOARD LEN)) (IMPLIES (AND (NOT (ZEROP BOARD)) (NOT (LESSP (REMAINDER BOARD 4) 3))) (p BOARD LEN))). Linear arithmetic, the lemmas COUNT-NUMBERP and QUOTIENT-4-DECREASES, and the definition of ZEROP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for LEN. The above induction scheme produces five new goals: Case 5. (IMPLIES (AND (ZEROP BOARD) (NUMBERP BOARD) (BOARDPN BOARD) (NOT (LESSP LEN (LENGTHN BOARD)))) (EQUAL (BOARD->NUMBER (NUMBER->BOARD BOARD LEN)) BOARD)), which simplifies, opening up ZEROP, NUMBERP, BOARDPN, LENGTHN, EQUAL, and LESSP, to: (IMPLIES (EQUAL BOARD 0) (EQUAL (BOARD->NUMBER (NUMBER->BOARD 0 LEN)) 0)). This again simplifies, trivially, to the new conjecture: (EQUAL (BOARD->NUMBER (NUMBER->BOARD 0 LEN)) 0), which we will name *1.1. Case 4. (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (NOT (BOARDPN (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN BOARD) (NOT (LESSP LEN (LENGTHN BOARD)))) (EQUAL (BOARD->NUMBER (NUMBER->BOARD BOARD LEN)) BOARD)). This simplifies, opening up ZEROP and BOARDPN, to: T. Case 3. (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (LESSP (SUB1 LEN) (LENGTHN (QUOTIENT BOARD 4))) (NUMBERP BOARD) (BOARDPN BOARD) (NOT (LESSP LEN (LENGTHN BOARD)))) (EQUAL (BOARD->NUMBER (NUMBER->BOARD BOARD LEN)) BOARD)). This simplifies, applying SUB1-ADD1, and unfolding the functions ZEROP, BOARDPN, LENGTHN, and LESSP, to: T. Case 2. (IMPLIES (AND (NOT (ZEROP BOARD)) (LESSP (REMAINDER BOARD 4) 3) (EQUAL (BOARD->NUMBER (NUMBER->BOARD (QUOTIENT BOARD 4) (SUB1 LEN))) (QUOTIENT BOARD 4)) (NUMBERP BOARD) (BOARDPN BOARD) (NOT (LESSP LEN (LENGTHN BOARD)))) (EQUAL (BOARD->NUMBER (NUMBER->BOARD BOARD LEN)) BOARD)), which simplifies, rewriting with SUB1-ADD1, PLUS-COMMUTES2, PLUS-COMMUTES1, CDR-CONS, and CAR-CONS, and expanding the functions ZEROP, BOARDPN, LENGTHN, LESSP, NUMBER->BOARD, NUMBER->PIECE, TIMES, EQUAL, NUMBERP, SUB1, PIECE->NUMBER, UNPACK, and BOARD->NUMBER, to the following three new conjectures: Case 2.3. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (EQUAL (BOARD->NUMBER (NUMBER->BOARD (QUOTIENT BOARD 4) (SUB1 LEN))) (QUOTIENT BOARD 4)) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (REMAINDER BOARD 4) 0)) (EQUAL (PLUS (QUOTIENT BOARD 4) (TIMES 3 (QUOTIENT BOARD 4)) 0) BOARD)). But this again simplifies, applying PLUS-COMMUTES1, and opening up the definitions of LESSP, EQUAL, and PLUS, to: (IMPLIES (AND (NOT (EQUAL BOARD 0)) (EQUAL (BOARD->NUMBER (NUMBER->BOARD (QUOTIENT BOARD 4) (SUB1 LEN))) (QUOTIENT BOARD 4)) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (REMAINDER BOARD 4) 0)) (EQUAL (PLUS (QUOTIENT BOARD 4) (TIMES 3 (QUOTIENT BOARD 4))) BOARD)). Applying the lemma REMAINDER-QUOTIENT-ELIM, replace BOARD by (PLUS Z (TIMES 4 X)) to eliminate (QUOTIENT BOARD 4) and (REMAINDER BOARD 4). We employ LESSP-REMAINDER, the type restriction lemma noted when QUOTIENT was introduced, and the type restriction lemma noted when REMAINDER was introduced to restrict the new variables. We would thus like to prove the new conjecture: (IMPLIES (AND (NUMBERP X) (NUMBERP Z) (EQUAL (LESSP Z 4) (NOT (ZEROP 4))) (NOT (EQUAL (PLUS Z (TIMES 4 X)) 0)) (EQUAL (BOARD->NUMBER (NUMBER->BOARD X (SUB1 LEN))) X) (BOARDPN X) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN X))) (EQUAL Z 0)) (EQUAL (PLUS X (TIMES 3 X)) (PLUS Z (TIMES 4 X)))), which further simplifies, using linear arithmetic, to: T. Case 2.2. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (EQUAL (BOARD->NUMBER (NUMBER->BOARD (QUOTIENT BOARD 4) (SUB1 LEN))) (QUOTIENT BOARD 4)) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN (QUOTIENT BOARD 4)))) (NOT (EQUAL (REMAINDER BOARD 4) 0)) (NOT (EQUAL (REMAINDER BOARD 4) 1))) (EQUAL (PLUS (QUOTIENT BOARD 4) (TIMES 3 (QUOTIENT BOARD 4)) 2) BOARD)), which again simplifies, using linear arithmetic, to: (IMPLIES (AND (EQUAL (REMAINDER BOARD 4) 2) (NOT (EQUAL BOARD 0)) (LESSP 2 3) (EQUAL (BOARD->NUMBER (NUMBER->BOARD (QUOTIENT BOARD 4) (SUB1 LEN))) (QUOTIENT BOARD 4)) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN (QUOTIENT BOARD 4)))) (NOT (EQUAL 2 0)) (NOT (EQUAL 2 1))) (EQUAL (PLUS (QUOTIENT BOARD 4) (TIMES 3 (QUOTIENT BOARD 4)) 2) BOARD)). But this again simplifies, rewriting with PLUS-COMMUTES1 and PLUS-COMMUTES2, and unfolding the definitions of LESSP and EQUAL, to: (IMPLIES (AND (EQUAL (REMAINDER BOARD 4) 2) (NOT (EQUAL BOARD 0)) (EQUAL (BOARD->NUMBER (NUMBER->BOARD (QUOTIENT BOARD 4) (SUB1 LEN))) (QUOTIENT BOARD 4)) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN (QUOTIENT BOARD 4))))) (EQUAL (PLUS 2 (QUOTIENT BOARD 4) (TIMES 3 (QUOTIENT BOARD 4))) BOARD)). Applying the lemma REMAINDER-QUOTIENT-ELIM, replace BOARD by (PLUS X (TIMES 4 Z)) to eliminate (REMAINDER BOARD 4) and (QUOTIENT BOARD 4). We rely upon LESSP-REMAINDER, the type restriction lemma noted when REMAINDER was introduced, and the type restriction lemma noted when QUOTIENT was introduced to restrict the new variables. This produces the new goal: (IMPLIES (AND (NUMBERP X) (EQUAL (LESSP X 4) (NOT (ZEROP 4))) (NUMBERP Z) (EQUAL X 2) (NOT (EQUAL (PLUS X (TIMES 4 Z)) 0)) (EQUAL (BOARD->NUMBER (NUMBER->BOARD Z (SUB1 LEN))) Z) (BOARDPN Z) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN Z)))) (EQUAL (PLUS 2 Z (TIMES 3 Z)) (PLUS X (TIMES 4 Z)))), which further simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (NOT (EQUAL BOARD 0)) (LESSP (REMAINDER BOARD 4) 3) (EQUAL (BOARD->NUMBER (NUMBER->BOARD (QUOTIENT BOARD 4) (SUB1 LEN))) (QUOTIENT BOARD 4)) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN (QUOTIENT BOARD 4)))) (NOT (EQUAL (REMAINDER BOARD 4) 0)) (EQUAL (REMAINDER BOARD 4) 1)) (EQUAL (PLUS (QUOTIENT BOARD 4) (TIMES 3 (QUOTIENT BOARD 4)) 1) BOARD)), which again simplifies, applying PLUS-COMMUTES1 and PLUS-COMMUTES2, and expanding the definitions of LESSP and EQUAL, to the new goal: (IMPLIES (AND (NOT (EQUAL BOARD 0)) (EQUAL (BOARD->NUMBER (NUMBER->BOARD (QUOTIENT BOARD 4) (SUB1 LEN))) (QUOTIENT BOARD 4)) (NUMBERP BOARD) (BOARDPN (QUOTIENT BOARD 4)) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (REMAINDER BOARD 4) 1)) (EQUAL (PLUS 1 (QUOTIENT BOARD 4) (TIMES 3 (QUOTIENT BOARD 4))) BOARD)). Applying the lemma REMAINDER-QUOTIENT-ELIM, replace BOARD by (PLUS Z (TIMES 4 X)) to eliminate (QUOTIENT BOARD 4) and (REMAINDER BOARD 4). We rely upon LESSP-REMAINDER, the type restriction lemma noted when QUOTIENT was introduced, and the type restriction lemma noted when REMAINDER was introduced to restrict the new variables. This produces the new conjecture: (IMPLIES (AND (NUMBERP X) (NUMBERP Z) (EQUAL (LESSP Z 4) (NOT (ZEROP 4))) (NOT (EQUAL (PLUS Z (TIMES 4 X)) 0)) (EQUAL (BOARD->NUMBER (NUMBER->BOARD X (SUB1 LEN))) X) (BOARDPN X) (NOT (EQUAL LEN 0)) (NUMBERP LEN) (NOT (LESSP (SUB1 LEN) (LENGTHN X))) (EQUAL Z 1)) (EQUAL (PLUS 1 X (TIMES 3 X)) (PLUS Z (TIMES 4 X)))), which further simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP BOARD)) (NOT (LESSP (REMAINDER BOARD 4) 3)) (NUMBERP BOARD) (BOARDPN BOARD) (NOT (LESSP LEN (LENGTHN BOARD)))) (EQUAL (BOARD->NUMBER (NUMBER->BOARD BOARD LEN)) BOARD)), which simplifies, unfolding the definitions of ZEROP and BOARDPN, to: T. So we now return to: (EQUAL (BOARD->NUMBER (NUMBER->BOARD 0 LEN)) 0), which we named *1.1 above. Let us appeal to the induction principle. There is only one suggested induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP LEN) (p LEN)) (IMPLIES (AND (NOT (ZEROP LEN)) (p (SUB1 LEN))) (p LEN))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to prove that the measure (COUNT LEN) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces the following two new goals: Case 2. (IMPLIES (ZEROP LEN) (EQUAL (BOARD->NUMBER (NUMBER->BOARD 0 LEN)) 0)). This simplifies, unfolding the functions ZEROP, NUMBER->BOARD, BOARD->NUMBER, and EQUAL, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP LEN)) (EQUAL (BOARD->NUMBER (NUMBER->BOARD 0 (SUB1 LEN))) 0)) (EQUAL (BOARD->NUMBER (NUMBER->BOARD 0 LEN)) 0)). This simplifies, applying CDR-CONS and CAR-CONS, and unfolding ZEROP, NUMBER->BOARD, REMAINDER, NUMBER->PIECE, QUOTIENT, PLUS, TIMES, PIECE->NUMBER, BOARD->NUMBER, and EQUAL, to: T. That finishes the proof of *1.1, which, consequently, finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.1 ] NUMBER->BOARD->NUMBER (PROVE-LEMMA LENGTHN-BOARD->NUMBER (REWRITE) (NOT (LESSP (LENGTH BOARD) (LENGTHN (BOARD->NUMBER BOARD))))) WARNING: Note that the proposed lemma LENGTHN-BOARD->NUMBER is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. Name the conjecture *1. Perhaps we can prove it by induction. There are two plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (NLISTP BOARD) (p BOARD)) (IMPLIES (AND (NOT (NLISTP BOARD)) (p (CDR BOARD))) (p BOARD))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP establish that the measure (COUNT BOARD) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme produces the following two new goals: Case 2. (IMPLIES (NLISTP BOARD) (NOT (LESSP (LENGTH BOARD) (LENGTHN (BOARD->NUMBER BOARD))))). This simplifies, expanding NLISTP, LENGTH, BOARD->NUMBER, LENGTHN, and LESSP, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP BOARD)) (NOT (LESSP (LENGTH (CDR BOARD)) (LENGTHN (BOARD->NUMBER (CDR BOARD)))))) (NOT (LESSP (LENGTH BOARD) (LENGTHN (BOARD->NUMBER BOARD))))). This simplifies, rewriting with PLUS-COMMUTES1 and SUB1-ADD1, and unfolding the definitions of NLISTP, LENGTH, BOARD->NUMBER, PIECE->NUMBER, and LESSP, to three new goals: Case 1.3. (IMPLIES (AND (LISTP BOARD) (NOT (LESSP (LENGTH (CDR BOARD)) (LENGTHN (BOARD->NUMBER (CDR BOARD))))) (CAR BOARD) (NOT (EQUAL (CAR BOARD) 'X)) (NOT (EQUAL (LENGTHN (PLUS (TIMES 4 (BOARD->NUMBER (CDR BOARD))) 2)) 0))) (NOT (LESSP (LENGTH (CDR BOARD)) (SUB1 (LENGTHN (PLUS (TIMES 4 (BOARD->NUMBER (CDR BOARD))) 2)))))), which again simplifies, applying PLUS-COMMUTES1, QUOTIENT-PLUS-TIMES, and SUB1-ADD1, and unfolding the functions LESSP and LENGTHN, to: T. Case 1.2. (IMPLIES (AND (LISTP BOARD) (NOT (LESSP (LENGTH (CDR BOARD)) (LENGTHN (BOARD->NUMBER (CDR BOARD))))) (CAR BOARD) (EQUAL (CAR BOARD) 'X) (NOT (EQUAL (LENGTHN (PLUS (TIMES 4 (BOARD->NUMBER (CDR BOARD))) 1)) 0))) (NOT (LESSP (LENGTH (CDR BOARD)) (SUB1 (LENGTHN (PLUS (TIMES 4 (BOARD->NUMBER (CDR BOARD))) 1)))))). However this again simplifies, applying PLUS-COMMUTES1, QUOTIENT-PLUS-TIMES, and SUB1-ADD1, and unfolding the functions LESSP and LENGTHN, to: T. Case 1.1. (IMPLIES (AND (LISTP BOARD) (NOT (LESSP (LENGTH (CDR BOARD)) (LENGTHN (BOARD->NUMBER (CDR BOARD))))) (NOT (CAR BOARD)) (NOT (EQUAL (LENGTHN (PLUS (TIMES 4 (BOARD->NUMBER (CDR BOARD))) 0)) 0))) (NOT (LESSP (LENGTH (CDR BOARD)) (SUB1 (LENGTHN (PLUS (TIMES 4 (BOARD->NUMBER (CDR BOARD))) 0)))))). However this again simplifies, rewriting with PLUS-COMMUTES1, QUOTIENT-PLUS-TIMES-INSTANCE, and SUB1-ADD1, and opening up the definitions of EQUAL, PLUS, and LENGTHN, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LENGTHN-BOARD->NUMBER (PROVE-LEMMA LENGTHN-MOVEN NIL (IMPLIES (OR (EQUAL X 1) (EQUAL X 2)) (EQUAL (LENGTHN (MOVEN X I BOARD)) (IF (LESSP I (LENGTHN BOARD)) (LENGTHN BOARD) (ADD1 I))))) This conjecture simplifies, opening up the function OR, to four new formulas: Case 4. (IMPLIES (AND (EQUAL X 1) (NOT (LESSP I (LENGTHN BOARD)))) (EQUAL (LENGTHN (MOVEN 1 I BOARD)) (ADD1 I))), which again simplifies, trivially, to the new goal: (IMPLIES (NOT (LESSP I (LENGTHN BOARD))) (EQUAL (LENGTHN (MOVEN 1 I BOARD)) (ADD1 I))), which we will name *1. Case 3. (IMPLIES (AND (EQUAL X 1) (LESSP I (LENGTHN BOARD))) (EQUAL (LENGTHN (MOVEN 1 I BOARD)) (LENGTHN BOARD))). This again simplifies, trivially, to: (IMPLIES (LESSP I (LENGTHN BOARD)) (EQUAL (LENGTHN (MOVEN 1 I BOARD)) (LENGTHN BOARD))), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (AND (IMPLIES (EQUAL X 2) (EQUAL (LENGTHN (MOVEN X I BOARD)) (IF (LESSP I (LENGTHN BOARD)) (LENGTHN BOARD) (ADD1 I)))) (IMPLIES (EQUAL X 1) (EQUAL (LENGTHN (MOVEN X I BOARD)) (IF (LESSP I (LENGTHN BOARD)) (LENGTHN BOARD) (ADD1 I))))). We named this *1. We will try to prove it by induction. The recursive terms in the conjecture suggest eight inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP I) (p X I BOARD)) (IMPLIES (AND (NOT (ZEROP I)) (p X (SUB1 I) (QUOTIENT BOARD 4))) (p X I BOARD))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP establish that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for BOARD. The above induction scheme leads to four new goals: Case 4. (IMPLIES (AND (ZEROP I) (EQUAL X 2)) (EQUAL (LENGTHN (MOVEN X I BOARD)) (IF (LESSP I (LENGTHN BOARD)) (LENGTHN BOARD) (ADD1 I)))), which simplifies, appealing to the lemmas QUOTIENT-PLUS-TIMES and SUB1-TYPE-RESTRICTION, and expanding the functions ZEROP, EQUAL, MOVEN, LESSP, LENGTHN, and ADD1, to 14 new goals: Case 4.14. (IMPLIES (AND (EQUAL I 0) (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))) 0)) (EQUAL 0 (ADD1 (LENGTHN (QUOTIENT BOARD 4))))), which again simplifies, using linear arithmetic, to: T. Case 4.13. (IMPLIES (AND (EQUAL I 0) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 0)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, using linear arithmetic, to: T. Case 4.12. (IMPLIES (AND (EQUAL I 0) (NOT (NUMBERP BOARD))) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, applying the lemma ADD1-EQUAL, and expanding the definition of NUMBERP, to the formula: (IMPLIES (NOT (NUMBERP BOARD)) (EQUAL (LENGTHN (QUOTIENT BOARD 4)) 0)). However this further simplifies, expanding the definitions of LESSP, NUMBERP, EQUAL, QUOTIENT, and LENGTHN, to: T. Case 4.11. (IMPLIES (AND (EQUAL I 0) (EQUAL BOARD 0)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, expanding QUOTIENT, LENGTHN, ADD1, and EQUAL, to: T. Case 4.10. (IMPLIES (AND (EQUAL I 0) (EQUAL BOARD 0)) (NOT (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 4.9. (IMPLIES (AND (EQUAL I 0) (NOT (NUMBERP BOARD))) (NOT (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 4.8. (IMPLIES (AND (EQUAL I 0) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 0)) (NOT (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 4.7. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))) 0)) (EQUAL 0 (ADD1 (LENGTHN (QUOTIENT BOARD 4))))), which again simplifies, using linear arithmetic, to: T. Case 4.6. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 0)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, using linear arithmetic, to: T. Case 4.5. (IMPLIES (AND (NOT (NUMBERP I)) (NOT (NUMBERP BOARD))) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, applying ADD1-EQUAL, and opening up the definition of NUMBERP, to: (IMPLIES (AND (NOT (NUMBERP I)) (NOT (NUMBERP BOARD))) (EQUAL (LENGTHN (QUOTIENT BOARD 4)) 0)), which further simplifies, opening up LESSP, NUMBERP, EQUAL, QUOTIENT, and LENGTHN, to: T. Case 4.4. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL BOARD 0)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, unfolding the definitions of QUOTIENT, LENGTHN, ADD1, and EQUAL, to: T. Case 4.3. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL BOARD 0)) (NOT (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 4.2. (IMPLIES (AND (NOT (NUMBERP I)) (NOT (NUMBERP BOARD))) (NOT (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 4.1. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 0)) (NOT (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (NOT (ZEROP I)) (EQUAL (LENGTHN (MOVEN X (SUB1 I) (QUOTIENT BOARD 4))) (IF (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4)) (ADD1 (SUB1 I)))) (EQUAL X 2)) (EQUAL (LENGTHN (MOVEN X I BOARD)) (IF (LESSP I (LENGTHN BOARD)) (LENGTHN BOARD) (ADD1 I)))), which simplifies, rewriting with ADD1-SUB1, and expanding the definitions of ZEROP, MOVEN, and LENGTHN, to the following 12 new goals: Case 3.12. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD) (NOT (LESSP I (ADD1 (LENGTHN (QUOTIENT BOARD 4)))))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))). But this again simplifies, rewriting with SUB1-ADD1, and expanding LESSP, to the new formula: (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))). Applying the lemma REMAINDER-QUOTIENT-ELIM, replace BOARD by (PLUS Z (TIMES 4 V)) to eliminate (REMAINDER BOARD 4) and (QUOTIENT BOARD 4). We use LESSP-REMAINDER, the type restriction lemma noted when REMAINDER was introduced, and the type restriction lemma noted when QUOTIENT was introduced to restrict the new variables. This produces the new conjecture: (IMPLIES (AND (NUMBERP Z) (EQUAL (LESSP Z 4) (NOT (ZEROP 4))) (NUMBERP V) (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN V))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) V)) I) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (EQUAL (LENGTHN (PLUS Z (TIMES 4 (MOVEN 2 (SUB1 I) V)))) (ADD1 I))), which further simplifies, appealing to the lemma QUOTIENT-PLUS-TIMES, and expanding the definitions of ZEROP, NOT, and LENGTHN, to: (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN V))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) V)) I) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0)) (EQUAL (PLUS Z (TIMES 4 (MOVEN 2 (SUB1 I) V))) 0)) (EQUAL 0 (ADD1 I))). However this again simplifies, using linear arithmetic, applying DIFFERENCE-IS-0 and SUB1-ADD1, and expanding the definitions of EQUAL, NUMBERP, QUOTIENT, ADD1, and LENGTHN, to the following two new goals: Case 3.12.2. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN V))) (NOT (EQUAL (MOVEN 2 (SUB1 I) V) 0)) (LESSP (MOVEN 2 (SUB1 I) V) 4) (EQUAL (ADD1 (LENGTHN 0)) I) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 2 (LENGTHN 0) V))) 0))). This again simplifies, rewriting with TIMES-DISTRIBUTES1 and PLUS-COMMUTES2, and expanding the functions LENGTHN, ADD1, EQUAL, SUB1, LESSP, MOVEN, and TIMES, to the new conjecture: (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (EQUAL (LENGTHN V) 0) (NOT (EQUAL (PLUS 2 (TIMES 4 (QUOTIENT V 4))) 0)) (LESSP (PLUS 2 (TIMES 4 (QUOTIENT V 4))) 4) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS 8 Z (TIMES 4 4 (QUOTIENT V 4))) 0))), which finally simplifies, using linear arithmetic, to: T. Case 3.12.1. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN V))) (NOT (EQUAL (MOVEN 2 (SUB1 I) V) 0)) (NOT (LESSP (MOVEN 2 (SUB1 I) V) 4)) (EQUAL (ADD1 (LENGTHN 1)) I) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 2 (LENGTHN 1) V))) 0))), which again simplifies, expanding the definitions of LENGTHN, ADD1, EQUAL, and SUB1, to: (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (LESSP 1 (LENGTHN V))) (NOT (EQUAL (MOVEN 2 1 V) 0)) (NOT (LESSP (MOVEN 2 1 V) 4)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 2 1 V))) 0))). This finally simplifies, using linear arithmetic, to: T. Case 3.11. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (NUMBERP BOARD)) (NOT (LESSP I 0))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, applying QUOTIENT-PLUS-TIMES-INSTANCE, and opening up the definitions of LESSP, NUMBERP, EQUAL, QUOTIENT, REMAINDER, PLUS, and LENGTHN, to the new conjecture: (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) 0)) I) (NOT (NUMBERP BOARD)) (EQUAL (TIMES 4 (MOVEN 2 (SUB1 I) 0)) 0)) (EQUAL 0 (ADD1 I))), which again simplifies, using linear arithmetic, rewriting with DIFFERENCE-IS-0 and SUB1-ADD1, and opening up the functions EQUAL, NUMBERP, QUOTIENT, ADD1, and LENGTHN, to the following two new formulas: Case 3.11.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (NOT (EQUAL (MOVEN 2 (SUB1 I) 0) 0)) (LESSP (MOVEN 2 (SUB1 I) 0) 4) (EQUAL (ADD1 (LENGTHN 0)) I) (NOT (NUMBERP BOARD))) (NOT (EQUAL (TIMES 4 (MOVEN 2 (LENGTHN 0) 0)) 0))). But this again simplifies, unfolding the functions LENGTHN, ADD1, EQUAL, SUB1, LESSP, MOVEN, and TIMES, to: T. Case 3.11.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (NOT (EQUAL (MOVEN 2 (SUB1 I) 0) 0)) (NOT (LESSP (MOVEN 2 (SUB1 I) 0) 4)) (EQUAL (ADD1 (LENGTHN 1)) I) (NOT (NUMBERP BOARD))) (NOT (EQUAL (TIMES 4 (MOVEN 2 (LENGTHN 1) 0)) 0))), which again simplifies, opening up the definitions of LENGTHN, ADD1, EQUAL, SUB1, MOVEN, LESSP, and TIMES, to: T. Case 3.10. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) I) (EQUAL BOARD 0) (NOT (LESSP I 0))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, rewriting with the lemma QUOTIENT-PLUS-TIMES-INSTANCE, and opening up the functions QUOTIENT, LENGTHN, EQUAL, LESSP, REMAINDER, and PLUS, to the conjecture: (IMPLIES (AND (NOT (EQUAL I 0)) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) 0)) I) (EQUAL (TIMES 4 (MOVEN 2 (SUB1 I) 0)) 0)) (EQUAL 0 (ADD1 I))). This again simplifies, using linear arithmetic, appealing to the lemmas DIFFERENCE-IS-0 and SUB1-ADD1, and unfolding EQUAL, NUMBERP, QUOTIENT, ADD1, and LENGTHN, to two new formulas: Case 3.10.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (EQUAL (MOVEN 2 (SUB1 I) 0) 0)) (LESSP (MOVEN 2 (SUB1 I) 0) 4) (EQUAL (ADD1 (LENGTHN 0)) I)) (NOT (EQUAL (TIMES 4 (MOVEN 2 (LENGTHN 0) 0)) 0))), which again simplifies, expanding the functions LENGTHN, ADD1, EQUAL, SUB1, MOVEN, LESSP, and TIMES, to: T. Case 3.10.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (EQUAL (MOVEN 2 (SUB1 I) 0) 0)) (NOT (LESSP (MOVEN 2 (SUB1 I) 0) 4)) (EQUAL (ADD1 (LENGTHN 1)) I)) (NOT (EQUAL (TIMES 4 (MOVEN 2 (LENGTHN 1) 0)) 0))), which again simplifies, opening up the definitions of LENGTHN, ADD1, EQUAL, SUB1, MOVEN, LESSP, and TIMES, to: T. Case 3.9. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (NUMBERP BOARD)) (LESSP I 0)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) 0)), which again simplifies, using linear arithmetic, to: T. Case 3.8. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) I) (EQUAL BOARD 0) (LESSP I 0)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) 0)), which again simplifies, using linear arithmetic, to: T. Case 3.7. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD) (LESSP I (ADD1 (LENGTHN (QUOTIENT BOARD 4))))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 (LENGTHN (QUOTIENT BOARD 4))))), which again simplifies, using linear arithmetic, to: T. Case 3.6. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD) (NOT (LESSP I (ADD1 (LENGTHN (QUOTIENT BOARD 4)))))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, using linear arithmetic, to: T. Case 3.5. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (NOT (LESSP I 0))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, rewriting with QUOTIENT-PLUS-TIMES-INSTANCE, and opening up the functions LESSP, NUMBERP, EQUAL, QUOTIENT, REMAINDER, PLUS, and LENGTHN, to the following two new goals: Case 3.5.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) 0)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (NOT (EQUAL (TIMES 4 (MOVEN 2 (SUB1 I) 0)) 0))) (EQUAL (ADD1 (LENGTHN (MOVEN 2 (SUB1 I) 0))) (ADD1 I))). This again simplifies, applying the lemma ADD1-EQUAL, to the conjecture: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) 0)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (NOT (EQUAL (TIMES 4 (MOVEN 2 (SUB1 I) 0)) 0))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) 0)) I)). But this further simplifies, expanding the definitions of LESSP, NUMBERP, EQUAL, QUOTIENT, and LENGTHN, to: T. Case 3.5.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) 0)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (EQUAL (TIMES 4 (MOVEN 2 (SUB1 I) 0)) 0)) (EQUAL 0 (ADD1 I))), which again simplifies, using linear arithmetic, applying DIFFERENCE-IS-0, and expanding EQUAL, NUMBERP, QUOTIENT, ADD1, LENGTHN, and TIMES, to the following three new formulas: Case 3.5.1.3. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL (MOVEN 2 (SUB1 I) 0) 0)) (LESSP (MOVEN 2 (SUB1 I) 0) 4) (EQUAL (ADD1 (LENGTHN 0)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD))) (NOT (EQUAL (TIMES 4 (MOVEN 2 (SUB1 I) 0)) 0))). This again simplifies, using linear arithmetic, to: T. Case 3.5.1.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL (MOVEN 2 (SUB1 I) 0) 0)) (NOT (LESSP (MOVEN 2 (SUB1 I) 0) 4)) (EQUAL (ADD1 (LENGTHN 1)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD))) (NOT (EQUAL (TIMES 4 (MOVEN 2 (SUB1 I) 0)) 0))), which again simplifies, using linear arithmetic, to: T. Case 3.5.1.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (MOVEN 2 (SUB1 I) 0) 0) (EQUAL 0 (LENGTHN (QUOTIENT BOARD 4)))) (NUMBERP BOARD)), which again simplifies, using linear arithmetic, to: T. Case 3.4. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL BOARD 0) (NOT (LESSP I 0))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, opening up the definitions of QUOTIENT, LENGTHN, EQUAL, and LESSP, to: T. Case 3.3. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (LESSP I 0)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) 0)), which again simplifies, using linear arithmetic, to: T. Case 3.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL BOARD 0) (LESSP I 0)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) 0)), which again simplifies, using linear arithmetic, to: T. Case 3.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD) (LESSP I (ADD1 (LENGTHN (QUOTIENT BOARD 4))))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 (LENGTHN (QUOTIENT BOARD 4))))), which again simplifies, appealing to the lemma SUB1-ADD1, and opening up the definition of LESSP, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 2 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 (LENGTHN (QUOTIENT BOARD 4))))). Appealing to the lemma REMAINDER-QUOTIENT-ELIM, we now replace BOARD by (PLUS Z (TIMES 4 V)) to eliminate (REMAINDER BOARD 4) and (QUOTIENT BOARD 4). We employ LESSP-REMAINDER, the type restriction lemma noted when REMAINDER was introduced, and the type restriction lemma noted when QUOTIENT was introduced to constrain the new variables. This generates: (IMPLIES (AND (NUMBERP Z) (EQUAL (LESSP Z 4) (NOT (ZEROP 4))) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) V)) (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (EQUAL (LENGTHN (PLUS Z (TIMES 4 (MOVEN 2 (SUB1 I) V)))) (ADD1 (LENGTHN V)))). But this further simplifies, appealing to the lemma QUOTIENT-PLUS-TIMES, and opening up the definitions of ZEROP, NOT, and LENGTHN, to the conjecture: (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (EQUAL (LENGTHN (MOVEN 2 (SUB1 I) V)) (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0)) (EQUAL (PLUS Z (TIMES 4 (MOVEN 2 (SUB1 I) V))) 0)) (EQUAL 0 (ADD1 (LENGTHN V)))). However this again simplifies, using linear arithmetic, rewriting with DIFFERENCE-IS-0 and PLUS-COMMUTES1, and opening up the definitions of EQUAL, NUMBERP, QUOTIENT, ADD1, LENGTHN, TIMES, and PLUS, to the following three new conjectures: Case 3.1.3. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (NOT (EQUAL (MOVEN 2 (SUB1 I) V) 0)) (LESSP (MOVEN 2 (SUB1 I) V) 4) (EQUAL (ADD1 (LENGTHN 0)) (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 2 (SUB1 I) V))) 0))). This finally simplifies, using linear arithmetic, to: T. Case 3.1.2. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (NOT (EQUAL (MOVEN 2 (SUB1 I) V) 0)) (NOT (LESSP (MOVEN 2 (SUB1 I) V) 4)) (EQUAL (ADD1 (LENGTHN 1)) (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 2 (SUB1 I) V))) 0))), which finally simplifies, using linear arithmetic, to: T. Case 3.1.1. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (EQUAL (MOVEN 2 (SUB1 I) V) 0) (EQUAL 0 (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL Z 0))), which finally simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (ZEROP I) (EQUAL X 1)) (EQUAL (LENGTHN (MOVEN X I BOARD)) (IF (LESSP I (LENGTHN BOARD)) (LENGTHN BOARD) (ADD1 I)))), which simplifies, applying QUOTIENT-PLUS-TIMES and SUB1-TYPE-RESTRICTION, and opening up the functions ZEROP, EQUAL, MOVEN, LESSP, LENGTHN, and ADD1, to the following 14 new formulas: Case 2.14. (IMPLIES (AND (EQUAL I 0) (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))) 0)) (EQUAL 0 (ADD1 (LENGTHN (QUOTIENT BOARD 4))))). However this again simplifies, using linear arithmetic, to: T. Case 2.13. (IMPLIES (AND (EQUAL I 0) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 0)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, using linear arithmetic, to: T. Case 2.12. (IMPLIES (AND (EQUAL I 0) (NOT (NUMBERP BOARD))) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, applying ADD1-EQUAL, and expanding the function NUMBERP, to the new formula: (IMPLIES (NOT (NUMBERP BOARD)) (EQUAL (LENGTHN (QUOTIENT BOARD 4)) 0)), which further simplifies, opening up the functions LESSP, NUMBERP, EQUAL, QUOTIENT, and LENGTHN, to: T. Case 2.11. (IMPLIES (AND (EQUAL I 0) (EQUAL BOARD 0)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, expanding QUOTIENT, LENGTHN, ADD1, and EQUAL, to: T. Case 2.10. (IMPLIES (AND (EQUAL I 0) (EQUAL BOARD 0)) (NOT (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 2.9. (IMPLIES (AND (EQUAL I 0) (NOT (NUMBERP BOARD))) (NOT (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 2.8. (IMPLIES (AND (EQUAL I 0) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 0)) (NOT (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 2.7. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))) 0)) (EQUAL 0 (ADD1 (LENGTHN (QUOTIENT BOARD 4))))), which again simplifies, using linear arithmetic, to: T. Case 2.6. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 0)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, using linear arithmetic, to: T. Case 2.5. (IMPLIES (AND (NOT (NUMBERP I)) (NOT (NUMBERP BOARD))) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, applying the lemma ADD1-EQUAL, and unfolding the function NUMBERP, to the formula: (IMPLIES (AND (NOT (NUMBERP I)) (NOT (NUMBERP BOARD))) (EQUAL (LENGTHN (QUOTIENT BOARD 4)) 0)). However this further simplifies, unfolding LESSP, NUMBERP, EQUAL, QUOTIENT, and LENGTHN, to: T. Case 2.4. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL BOARD 0)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 1)), which again simplifies, unfolding QUOTIENT, LENGTHN, ADD1, and EQUAL, to: T. Case 2.3. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL BOARD 0)) (NOT (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 2.2. (IMPLIES (AND (NOT (NUMBERP I)) (NOT (NUMBERP BOARD))) (NOT (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL (ADD1 (LENGTHN (QUOTIENT BOARD 4))) 0)) (NOT (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT BOARD 4))) 0))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP I)) (EQUAL (LENGTHN (MOVEN X (SUB1 I) (QUOTIENT BOARD 4))) (IF (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4)) (ADD1 (SUB1 I)))) (EQUAL X 1)) (EQUAL (LENGTHN (MOVEN X I BOARD)) (IF (LESSP I (LENGTHN BOARD)) (LENGTHN BOARD) (ADD1 I)))), which simplifies, rewriting with the lemma ADD1-SUB1, and expanding the functions ZEROP, MOVEN, and LENGTHN, to 12 new conjectures: Case 1.12. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD) (NOT (LESSP I (ADD1 (LENGTHN (QUOTIENT BOARD 4)))))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, applying SUB1-ADD1, and unfolding LESSP, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))). Applying the lemma REMAINDER-QUOTIENT-ELIM, replace BOARD by (PLUS Z (TIMES 4 V)) to eliminate (REMAINDER BOARD 4) and (QUOTIENT BOARD 4). We rely upon LESSP-REMAINDER, the type restriction lemma noted when REMAINDER was introduced, and the type restriction lemma noted when QUOTIENT was introduced to restrict the new variables. This produces: (IMPLIES (AND (NUMBERP Z) (EQUAL (LESSP Z 4) (NOT (ZEROP 4))) (NUMBERP V) (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN V))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) V)) I) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (EQUAL (LENGTHN (PLUS Z (TIMES 4 (MOVEN 1 (SUB1 I) V)))) (ADD1 I))), which further simplifies, applying QUOTIENT-PLUS-TIMES, and opening up the definitions of ZEROP, NOT, and LENGTHN, to the new goal: (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN V))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) V)) I) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0)) (EQUAL (PLUS Z (TIMES 4 (MOVEN 1 (SUB1 I) V))) 0)) (EQUAL 0 (ADD1 I))), which again simplifies, using linear arithmetic, applying DIFFERENCE-IS-0 and SUB1-ADD1, and unfolding EQUAL, NUMBERP, QUOTIENT, ADD1, and LENGTHN, to the following two new conjectures: Case 1.12.2. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN V))) (NOT (EQUAL (MOVEN 1 (SUB1 I) V) 0)) (LESSP (MOVEN 1 (SUB1 I) V) 4) (EQUAL (ADD1 (LENGTHN 0)) I) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 1 (LENGTHN 0) V))) 0))). This again simplifies, appealing to the lemmas TIMES-DISTRIBUTES1 and PLUS-COMMUTES2, and opening up the definitions of LENGTHN, ADD1, EQUAL, SUB1, LESSP, MOVEN, and TIMES, to the goal: (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (EQUAL (LENGTHN V) 0) (NOT (EQUAL (PLUS 1 (TIMES 4 (QUOTIENT V 4))) 0)) (LESSP (PLUS 1 (TIMES 4 (QUOTIENT V 4))) 4) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS 4 Z (TIMES 4 4 (QUOTIENT V 4))) 0))). This finally simplifies, using linear arithmetic, to: T. Case 1.12.1. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN V))) (NOT (EQUAL (MOVEN 1 (SUB1 I) V) 0)) (NOT (LESSP (MOVEN 1 (SUB1 I) V) 4)) (EQUAL (ADD1 (LENGTHN 1)) I) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 1 (LENGTHN 1) V))) 0))), which again simplifies, expanding the functions LENGTHN, ADD1, EQUAL, and SUB1, to: (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (LESSP 1 (LENGTHN V))) (NOT (EQUAL (MOVEN 1 1 V) 0)) (NOT (LESSP (MOVEN 1 1 V) 4)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 1 1 V))) 0))). But this finally simplifies, using linear arithmetic, to: T. Case 1.11. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (NUMBERP BOARD)) (NOT (LESSP I 0))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, rewriting with the lemma QUOTIENT-PLUS-TIMES-INSTANCE, and unfolding the functions LESSP, NUMBERP, EQUAL, QUOTIENT, REMAINDER, PLUS, and LENGTHN, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) 0)) I) (NOT (NUMBERP BOARD)) (EQUAL (TIMES 4 (MOVEN 1 (SUB1 I) 0)) 0)) (EQUAL 0 (ADD1 I))). However this again simplifies, using linear arithmetic, applying DIFFERENCE-IS-0 and SUB1-ADD1, and expanding the definitions of EQUAL, NUMBERP, QUOTIENT, ADD1, and LENGTHN, to the following two new conjectures: Case 1.11.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (NOT (EQUAL (MOVEN 1 (SUB1 I) 0) 0)) (LESSP (MOVEN 1 (SUB1 I) 0) 4) (EQUAL (ADD1 (LENGTHN 0)) I) (NOT (NUMBERP BOARD))) (NOT (EQUAL (TIMES 4 (MOVEN 1 (LENGTHN 0) 0)) 0))). This again simplifies, expanding LENGTHN, ADD1, EQUAL, SUB1, LESSP, MOVEN, and TIMES, to: T. Case 1.11.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (NOT (EQUAL (MOVEN 1 (SUB1 I) 0) 0)) (NOT (LESSP (MOVEN 1 (SUB1 I) 0) 4)) (EQUAL (ADD1 (LENGTHN 1)) I) (NOT (NUMBERP BOARD))) (NOT (EQUAL (TIMES 4 (MOVEN 1 (LENGTHN 1) 0)) 0))), which again simplifies, opening up the functions LENGTHN, ADD1, EQUAL, SUB1, MOVEN, LESSP, and TIMES, to: T. Case 1.10. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) I) (EQUAL BOARD 0) (NOT (LESSP I 0))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, applying the lemma QUOTIENT-PLUS-TIMES-INSTANCE, and expanding the definitions of QUOTIENT, LENGTHN, EQUAL, LESSP, REMAINDER, and PLUS, to the formula: (IMPLIES (AND (NOT (EQUAL I 0)) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) 0)) I) (EQUAL (TIMES 4 (MOVEN 1 (SUB1 I) 0)) 0)) (EQUAL 0 (ADD1 I))). But this again simplifies, using linear arithmetic, applying DIFFERENCE-IS-0 and SUB1-ADD1, and expanding EQUAL, NUMBERP, QUOTIENT, ADD1, and LENGTHN, to the following two new formulas: Case 1.10.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (EQUAL (MOVEN 1 (SUB1 I) 0) 0)) (LESSP (MOVEN 1 (SUB1 I) 0) 4) (EQUAL (ADD1 (LENGTHN 0)) I)) (NOT (EQUAL (TIMES 4 (MOVEN 1 (LENGTHN 0) 0)) 0))). However this again simplifies, unfolding the functions LENGTHN, ADD1, EQUAL, SUB1, MOVEN, LESSP, and TIMES, to: T. Case 1.10.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NOT (EQUAL (MOVEN 1 (SUB1 I) 0) 0)) (NOT (LESSP (MOVEN 1 (SUB1 I) 0) 4)) (EQUAL (ADD1 (LENGTHN 1)) I)) (NOT (EQUAL (TIMES 4 (MOVEN 1 (LENGTHN 1) 0)) 0))), which again simplifies, expanding the definitions of LENGTHN, ADD1, EQUAL, SUB1, MOVEN, LESSP, and TIMES, to: T. Case 1.9. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (NUMBERP BOARD)) (LESSP I 0)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) 0)), which again simplifies, using linear arithmetic, to: T. Case 1.8. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) I) (EQUAL BOARD 0) (LESSP I 0)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) 0)), which again simplifies, using linear arithmetic, to: T. Case 1.7. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (NOT (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4)))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) I) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD) (LESSP I (ADD1 (LENGTHN (QUOTIENT BOARD 4))))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 (LENGTHN (QUOTIENT BOARD 4))))), which again simplifies, using linear arithmetic, to: T. Case 1.6. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD) (NOT (LESSP I (ADD1 (LENGTHN (QUOTIENT BOARD 4)))))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, using linear arithmetic, to: T. Case 1.5. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (NOT (LESSP I 0))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, appealing to the lemma QUOTIENT-PLUS-TIMES-INSTANCE, and unfolding LESSP, NUMBERP, EQUAL, QUOTIENT, REMAINDER, PLUS, and LENGTHN, to two new formulas: Case 1.5.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) 0)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (NOT (EQUAL (TIMES 4 (MOVEN 1 (SUB1 I) 0)) 0))) (EQUAL (ADD1 (LENGTHN (MOVEN 1 (SUB1 I) 0))) (ADD1 I))), which again simplifies, applying ADD1-EQUAL, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) 0)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (NOT (EQUAL (TIMES 4 (MOVEN 1 (SUB1 I) 0)) 0))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) 0)) I)), which further simplifies, expanding LESSP, NUMBERP, EQUAL, QUOTIENT, and LENGTHN, to: T. Case 1.5.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) 0)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (EQUAL (TIMES 4 (MOVEN 1 (SUB1 I) 0)) 0)) (EQUAL 0 (ADD1 I))), which again simplifies, using linear arithmetic, rewriting with DIFFERENCE-IS-0, and expanding the functions EQUAL, NUMBERP, QUOTIENT, ADD1, LENGTHN, and TIMES, to the following three new formulas: Case 1.5.1.3. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL (MOVEN 1 (SUB1 I) 0) 0)) (LESSP (MOVEN 1 (SUB1 I) 0) 4) (EQUAL (ADD1 (LENGTHN 0)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD))) (NOT (EQUAL (TIMES 4 (MOVEN 1 (SUB1 I) 0)) 0))). This again simplifies, using linear arithmetic, to: T. Case 1.5.1.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL (MOVEN 1 (SUB1 I) 0) 0)) (NOT (LESSP (MOVEN 1 (SUB1 I) 0) 4)) (EQUAL (ADD1 (LENGTHN 1)) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD))) (NOT (EQUAL (TIMES 4 (MOVEN 1 (SUB1 I) 0)) 0))), which again simplifies, using linear arithmetic, to: T. Case 1.5.1.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (MOVEN 1 (SUB1 I) 0) 0) (EQUAL 0 (LENGTHN (QUOTIENT BOARD 4)))) (NUMBERP BOARD)), which again simplifies, using linear arithmetic, to: T. Case 1.4. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL BOARD 0) (NOT (LESSP I 0))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 I))), which again simplifies, expanding the functions QUOTIENT, LENGTHN, EQUAL, and LESSP, to: T. Case 1.3. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (NUMBERP BOARD)) (LESSP I 0)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) 0)), which again simplifies, using linear arithmetic, to: T. Case 1.2. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL BOARD 0) (LESSP I 0)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) 0)), which again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD) (LESSP I (ADD1 (LENGTHN (QUOTIENT BOARD 4))))) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 (LENGTHN (QUOTIENT BOARD 4))))), which again simplifies, applying the lemma SUB1-ADD1, and opening up the definition of LESSP, to: (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN (QUOTIENT BOARD 4))) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))) (LENGTHN (QUOTIENT BOARD 4))) (NOT (EQUAL BOARD 0)) (NUMBERP BOARD)) (EQUAL (LENGTHN (PLUS (REMAINDER BOARD 4) (TIMES 4 (MOVEN 1 (SUB1 I) (QUOTIENT BOARD 4))))) (ADD1 (LENGTHN (QUOTIENT BOARD 4))))). Appealing to the lemma REMAINDER-QUOTIENT-ELIM, we now replace BOARD by (PLUS Z (TIMES 4 V)) to eliminate (REMAINDER BOARD 4) and (QUOTIENT BOARD 4). We rely upon LESSP-REMAINDER, the type restriction lemma noted when REMAINDER was introduced, and the type restriction lemma noted when QUOTIENT was introduced to constrain the new variables. This generates: (IMPLIES (AND (NUMBERP Z) (EQUAL (LESSP Z 4) (NOT (ZEROP 4))) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) V)) (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (EQUAL (LENGTHN (PLUS Z (TIMES 4 (MOVEN 1 (SUB1 I) V)))) (ADD1 (LENGTHN V)))). This further simplifies, rewriting with QUOTIENT-PLUS-TIMES, and expanding the functions ZEROP, NOT, and LENGTHN, to the new goal: (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (EQUAL (LENGTHN (MOVEN 1 (SUB1 I) V)) (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0)) (EQUAL (PLUS Z (TIMES 4 (MOVEN 1 (SUB1 I) V))) 0)) (EQUAL 0 (ADD1 (LENGTHN V)))), which again simplifies, using linear arithmetic, applying DIFFERENCE-IS-0 and PLUS-COMMUTES1, and opening up the functions EQUAL, NUMBERP, QUOTIENT, ADD1, LENGTHN, TIMES, and PLUS, to the following three new goals: Case 1.1.3. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (NOT (EQUAL (MOVEN 1 (SUB1 I) V) 0)) (LESSP (MOVEN 1 (SUB1 I) V) 4) (EQUAL (ADD1 (LENGTHN 0)) (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 1 (SUB1 I) V))) 0))). However this finally simplifies, using linear arithmetic, to: T. Case 1.1.2. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (NOT (EQUAL (MOVEN 1 (SUB1 I) V) 0)) (NOT (LESSP (MOVEN 1 (SUB1 I) V) 4)) (EQUAL (ADD1 (LENGTHN 1)) (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL (PLUS Z (TIMES 4 (MOVEN 1 (SUB1 I) V))) 0))), which finally simplifies, using linear arithmetic, to: T. Case 1.1.1. (IMPLIES (AND (NUMBERP Z) (LESSP Z 4) (NUMBERP V) (NOT (EQUAL I 0)) (NUMBERP I) (LESSP (SUB1 I) (LENGTHN V)) (EQUAL (MOVEN 1 (SUB1 I) V) 0) (EQUAL 0 (LENGTHN V)) (NOT (EQUAL (PLUS Z (TIMES 4 V)) 0))) (NOT (EQUAL Z 0))), which finally simplifies, using linear arithmetic, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.8 0.1 ] LENGTHN-MOVEN (PROVE-LEMMA LENGTHN-MOVEN-HACK-1 (REWRITE) (IMPLIES (AND (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (NOT (LESSP 9 (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD)))))) ((USE (LENGTHN-MOVEN (X 1) (BOARD (BOARD->NUMBER BOARD)))))) WARNING: Note that the proposed lemma LENGTHN-MOVEN-HACK-1 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This simplifies, applying SUB1-ADD1, and expanding the functions EQUAL, OR, IMPLIES, SUB1, NUMBERP, and LESSP, to the following three new conjectures: Case 3. (IMPLIES (AND (NOT (LESSP I (LENGTHN (BOARD->NUMBER BOARD)))) (EQUAL (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD))) (ADD1 I)) (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (NOT (NUMBERP I))) (NOT (LESSP 8 0))). This again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (NOT (LESSP I (LENGTHN (BOARD->NUMBER BOARD)))) (EQUAL (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD))) (ADD1 I)) (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (NUMBERP I)) (NOT (LESSP 8 I))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP I (LENGTHN (BOARD->NUMBER BOARD))) (EQUAL (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD))) (LENGTHN (BOARD->NUMBER BOARD))) (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (NOT (LESSP 9 (LENGTHN (BOARD->NUMBER BOARD))))), which again simplifies, using linear arithmetic and appealing to the lemma LENGTHN-BOARD->NUMBER, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LENGTHN-MOVEN-HACK-1 (PROVE-LEMMA LESSP-PICK-MOVEN (REWRITE) (IMPLIES (AND (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (LESSP (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) 9)) ((DISABLE LESSP-PICK-MOVE) (USE (LESSP-PICK-MOVE (BOARD (MOVE 'X I BOARD)))))) WARNING: Note that the proposed lemma LESSP-PICK-MOVEN is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations AND, IMPLIES, and LENGTH-MOVE, to: (IMPLIES (AND (IMPLIES (EQUAL (LENGTH BOARD) 9) (LESSP (PICK-MOVE (MOVE 'X I BOARD)) 9)) (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (LESSP (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) 9)), which simplifies, using linear arithmetic, applying MOVE-IS-MOVEN, BOARDP-NUMBER-BOARD, LENGTH-NUMBER->BOARD, LENGTHN-MOVEN-HACK-1, BOARDPN-MOVEN, BOARDPN-BOARD->NUMBER, NUMBER->BOARD->NUMBER, and PICK-MOVE-IS-PICK-MOVEN, and expanding the functions EQUAL, OR, PIECE->NUMBER, NUMBERP, and IMPLIES, to: T. Q.E.D. [ 0.0 0.0 0.0 ] LESSP-PICK-MOVEN (PROVE-LEMMA LENGTHN-MOVEN-HACK-2 (REWRITE) (IMPLIES (AND (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (NOT (LESSP 9 (LENGTHN (MOVEN 2 (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) (MOVEN 1 I (BOARD->NUMBER BOARD))))))) ((USE (LENGTHN-MOVEN (X 2) (I (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD)))) (BOARD (MOVEN 1 I (BOARD->NUMBER BOARD))))))) WARNING: Note that the proposed lemma LENGTHN-MOVEN-HACK-2 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This simplifies, rewriting with the lemma SUB1-ADD1, and unfolding the definitions of EQUAL, OR, IMPLIES, SUB1, NUMBERP, and LESSP, to three new goals: Case 3. (IMPLIES (AND (NOT (LESSP (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD))))) (EQUAL (LENGTHN (MOVEN 2 (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) (MOVEN 1 I (BOARD->NUMBER BOARD)))) (ADD1 (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))))) (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (NOT (NUMBERP (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD)))))) (NOT (LESSP 8 0))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (NOT (LESSP (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD))))) (EQUAL (LENGTHN (MOVEN 2 (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) (MOVEN 1 I (BOARD->NUMBER BOARD)))) (ADD1 (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))))) (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9) (NUMBERP (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))))) (NOT (LESSP 8 (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD)))))), which again simplifies, using linear arithmetic and rewriting with LESSP-PICK-MOVEN, to: T. Case 1. (IMPLIES (AND (LESSP (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD)))) (EQUAL (LENGTHN (MOVEN 2 (PICK-MOVEN (MOVEN 1 I (BOARD->NUMBER BOARD))) (MOVEN 1 I (BOARD->NUMBER BOARD)))) (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD)))) (LESSP I 9) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (NOT (LESSP 9 (LENGTHN (MOVEN 1 I (BOARD->NUMBER BOARD)))))). However this again simplifies, using linear arithmetic and rewriting with the lemma LENGTHN-MOVEN-HACK-1, to: T. Q.E.D. [ 0.0 0.1 0.0 ] LENGTHN-MOVEN-HACK-2 (PROVE-LEMMA LEMMA-B1 (REWRITE) (IMPLIES (AND (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1-A XMOVES BOARD I) (NUMBER->BOARD* (TIC-TAC-TOE1-B XMOVES (BOARD->NUMBER BOARD) I) 9))) ((INDUCT (TIC-TAC-TOE1-A XMOVES BOARD I)) (EXPAND (TIC-TAC-TOE1-A XMOVES BOARD I)) (DISABLE MOVE MOVEN LEGALP LEGALPN WINP WINPN BOARD->NUMBER NUMBER->BOARD PIECE->NUMBER NUMBER->PIECE))) This formula can be simplified, using the abbreviations IMPLIES, NOT, OR, AND, and LENGTH-MOVE, to the following four new goals: Case 4. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (LESSP I 9) (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1-A XMOVES BOARD I) (NUMBER->BOARD* (TIC-TAC-TOE1-B XMOVES (BOARD->NUMBER BOARD) I) 9))). This simplifies, using linear arithmetic, applying the lemmas LEGALP-IS-LEGALPN, MOVE-IS-MOVEN, BOARDP-NUMBER-BOARD, LENGTH-NUMBER->BOARD, LEGALPN-IMPLIES-LESSP, LENGTHN-MOVEN-HACK-1, BOARDPN-MOVEN, BOARDPN-BOARD->NUMBER, NUMBER->BOARD->NUMBER, PICK-MOVE-IS-PICK-MOVEN, LENGTHN-MOVEN-HACK-2, WINP-IS-WINPN, CDR-CONS, and CAR-CONS, and unfolding the definitions of OR, EQUAL, PIECE->NUMBER, NUMBERP, TIC-TAC-TOE1-A, TIC-TAC-TOE1-B, and NUMBER->BOARD*, to: T. Case 3. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINP 'O (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)))) (IMPLIES (AND (BOARDP (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1-A (CDR XMOVES) (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD)) (ADD1 (ADD1 I))) (NUMBER->BOARD* (TIC-TAC-TOE1-B (CDR XMOVES) (BOARD->NUMBER (MOVE 'O (PICK-MOVE (MOVE 'X (CAR XMOVES) BOARD)) (MOVE 'X (CAR XMOVES) BOARD))) (ADD1 (ADD1 I))) 9))) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1-A XMOVES BOARD I) (NUMBER->BOARD* (TIC-TAC-TOE1-B XMOVES (BOARD->NUMBER BOARD) I) 9))). This simplifies, using linear arithmetic, applying LEGALP-IS-LEGALPN, MOVE-IS-MOVEN, BOARDP-NUMBER-BOARD, LENGTH-NUMBER->BOARD, LEGALPN-IMPLIES-LESSP, LENGTHN-MOVEN-HACK-1, BOARDPN-MOVEN, BOARDPN-BOARD->NUMBER, NUMBER->BOARD->NUMBER, PICK-MOVE-IS-PICK-MOVEN, LENGTHN-MOVEN-HACK-2, and WINP-IS-WINPN, and expanding OR, EQUAL, PIECE->NUMBER, NUMBERP, AND, IMPLIES, and TIC-TAC-TOE1-A, to: (IMPLIES (AND (LEGALPN (CAR XMOVES) (BOARD->NUMBER BOARD)) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))))) (EQUAL (TIC-TAC-TOE1-A (CDR XMOVES) (NUMBER->BOARD (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) 9) (ADD1 (ADD1 I))) (NUMBER->BOARD* (TIC-TAC-TOE1-B (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) (ADD1 (ADD1 I))) 9)) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (EQUAL (CONS (NUMBER->BOARD (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD)) 9) (CONS (NUMBER->BOARD (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) 9) (NUMBER->BOARD* (TIC-TAC-TOE1-B (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) (MOVEN 1 (CAR XMOVES) (BOARD->NUMBER BOARD))) (ADD1 (ADD1 I))) 9))) (NUMBER->BOARD* (TIC-TAC-TOE1-B XMOVES (BOARD->NUMBER BOARD) I) 9))). However this again simplifies, applying CDR-CONS and CAR-CONS, and expanding the functions TIC-TAC-TOE1-B and NUMBER->BOARD*, to: T. Case 2. (IMPLIES (AND (LEGALP (CAR XMOVES) BOARD) (NOT (LESSP I 9)) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1-A XMOVES BOARD I) (NUMBER->BOARD* (TIC-TAC-TOE1-B XMOVES (BOARD->NUMBER BOARD) I) 9))). This simplifies, applying the lemmas LEGALP-IS-LEGALPN, MOVE-IS-MOVEN, CDR-CONS, and CAR-CONS, and expanding the definitions of TIC-TAC-TOE1-A, OR, EQUAL, PIECE->NUMBER, TIC-TAC-TOE1-B, and NUMBER->BOARD*, to: T. Case 1. (IMPLIES (AND (NOT (LEGALP (CAR XMOVES) BOARD)) (BOARDP BOARD) (EQUAL (LENGTH BOARD) 9)) (EQUAL (TIC-TAC-TOE1-A XMOVES BOARD I) (NUMBER->BOARD* (TIC-TAC-TOE1-B XMOVES (BOARD->NUMBER BOARD) I) 9))). This simplifies, applying LEGALP-IS-LEGALPN, MOVE-IS-MOVEN, CDR-CONS, and CAR-CONS, and unfolding TIC-TAC-TOE1-A, OR, EQUAL, PIECE->NUMBER, TIC-TAC-TOE1-B, and NUMBER->BOARD*, to: T. Q.E.D. [ 0.0 0.4 0.0 ] LEMMA-B1 (PROVE-LEMMA LEMMA-B (REWRITE) (EQUAL (TIC-TAC-TOE-A XMOVES) (NUMBER->BOARD* (TIC-TAC-TOE-B XMOVES) 9))) WARNING: Note that the rewrite rule LEMMA-B will be stored so as to apply only to terms with the nonrecursive function symbol TIC-TAC-TOE-A. This conjecture can be simplified, using the abbreviations TIC-TAC-TOE-B and TIC-TAC-TOE-A, to the conjecture: (EQUAL (CONS (LIST F F F F F F F F F) (TIC-TAC-TOE1-A XMOVES (LIST F F F F F F F F F) 1)) (NUMBER->BOARD* (CONS 0 (TIC-TAC-TOE1-B XMOVES 0 1)) 9)). This simplifies, applying LEMMA-B1, CDR-CONS, and CAR-CONS, and unfolding the definitions of EQUAL, LENGTH, BOARDP, BOARD->NUMBER, NUMBER->BOARD, and NUMBER->BOARD*, to: T. Q.E.D. [ 0.0 0.2 0.0 ] LEMMA-B (DEFN ARRAY->LST (I ARRAY) (COND ((NOT (LESSP I (LENGTH ARRAY))) NIL) ((NUMBERP (NTH I ARRAY)) (CONS (NTH I ARRAY) (ARRAY->LST (ADD1 I) ARRAY))) (T (LIST (NTH I ARRAY)))) ((LESSP (DIFFERENCE (LENGTH ARRAY) I)))) Linear arithmetic establishes that the measure: (DIFFERENCE (LENGTH ARRAY) I) decreases according to the well-founded relation LESSP in each recursive call. Hence, ARRAY->LST is accepted under the principle of definition. From the definition we can conclude that: (OR (LITATOM (ARRAY->LST I ARRAY)) (LISTP (ARRAY->LST I ARRAY))) is a theorem. [ 0.0 0.0 0.0 ] ARRAY->LST (DEFN PUT (VAL N LST) (IF (ZEROP N) (IF (LISTP LST) (CONS VAL (CDR LST)) (LIST VAL)) (CONS (CAR LST) (PUT VAL (SUB1 N) (CDR LST))))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be used to prove that the measure (COUNT N) decreases according to the well-founded relation LESSP in each recursive call. Hence, PUT is accepted under the principle of definition. Observe that (LISTP (PUT VAL N LST)) is a theorem. [ 0.0 0.0 0.0 ] PUT (PROVE-LEMMA NTH-PUT (REWRITE) (IMPLIES (AND (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY))) (EQUAL (NTH I (PUT VAL J ARRAY)) (IF (EQUAL I J) VAL (NTH I ARRAY))))) This conjecture simplifies, trivially, to the following two new conjectures: Case 2. (IMPLIES (AND (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY)) (NOT (EQUAL I J))) (EQUAL (NTH I (PUT VAL J ARRAY)) (NTH I ARRAY))). Call the above conjecture *1. Case 1. (IMPLIES (AND (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY)) (EQUAL I J)) (EQUAL (NTH I (PUT VAL J ARRAY)) VAL)). This again simplifies, obviously, to the new goal: (IMPLIES (AND (NUMBERP J) (LESSP J (LENGTH ARRAY))) (EQUAL (NTH J (PUT VAL J ARRAY)) VAL)), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (IMPLIES (AND (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY))) (EQUAL (NTH I (PUT VAL J ARRAY)) (IF (EQUAL I J) VAL (NTH I ARRAY)))). We named this *1. We will try to prove it by induction. The recursive terms in the conjecture suggest five inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY)))) (p I VAL J ARRAY)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (OR (EQUAL J 0) (NOT (NUMBERP J)))) (p I VAL J ARRAY)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (NOT (OR (EQUAL J 0) (NOT (NUMBERP J)))) (p (SUB1 I) VAL (SUB1 J) (CDR ARRAY))) (p I VAL J ARRAY))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT establish that the measure (COUNT J) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instances chosen for I and ARRAY. The above induction scheme leads to four new formulas: Case 4. (IMPLIES (AND (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY)))) (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY))) (EQUAL (NTH I (PUT VAL J ARRAY)) (IF (EQUAL I J) VAL (NTH I ARRAY)))), which simplifies, unfolding the functions LENGTH, NOT, OR, EQUAL, and LESSP, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (OR (EQUAL J 0) (NOT (NUMBERP J))) (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY))) (EQUAL (NTH I (PUT VAL J ARRAY)) (IF (EQUAL I J) VAL (NTH I ARRAY)))), which simplifies, opening up the functions LENGTH, NOT, OR, NUMBERP, EQUAL, LESSP, PUT, and NTH, to two new goals: Case 3.2. (IMPLIES (AND (LISTP ARRAY) (NOT (EQUAL (ADD1 (LENGTH (CDR ARRAY))) 0)) (EQUAL J 0) (NUMBERP I) (NOT (EQUAL I 0))) (EQUAL (NTH I (CONS VAL (CDR ARRAY))) (NTH (SUB1 I) (CDR ARRAY)))), which again simplifies, rewriting with CDR-CONS, and opening up the function NTH, to: T. Case 3.1. (IMPLIES (AND (LISTP ARRAY) (NOT (EQUAL (ADD1 (LENGTH (CDR ARRAY))) 0)) (EQUAL J 0) (NUMBERP I) (EQUAL I 0)) (EQUAL (NTH I (CONS VAL (CDR ARRAY))) VAL)). However this again simplifies, applying the lemma CAR-CONS, and expanding the definitions of NUMBERP, EQUAL, and NTH, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (NOT (OR (EQUAL J 0) (NOT (NUMBERP J)))) (NOT (LESSP (SUB1 J) (LENGTH (CDR ARRAY)))) (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY))) (EQUAL (NTH I (PUT VAL J ARRAY)) (IF (EQUAL I J) VAL (NTH I ARRAY)))), which simplifies, rewriting with SUB1-ADD1, and unfolding LENGTH, NOT, OR, and LESSP, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (NOT (OR (EQUAL J 0) (NOT (NUMBERP J)))) (EQUAL (NTH (SUB1 I) (PUT VAL (SUB1 J) (CDR ARRAY))) (IF (EQUAL (SUB1 I) (SUB1 J)) VAL (NTH (SUB1 I) (CDR ARRAY)))) (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY))) (EQUAL (NTH I (PUT VAL J ARRAY)) (IF (EQUAL I J) VAL (NTH I ARRAY)))). This simplifies, applying SUB1-ADD1, CDR-CONS, and CAR-CONS, and opening up LENGTH, NOT, OR, LESSP, PUT, and NTH, to four new formulas: Case 1.4. (IMPLIES (AND (LISTP ARRAY) (NOT (EQUAL (ADD1 (LENGTH (CDR ARRAY))) 0)) (NOT (EQUAL J 0)) (NOT (EQUAL (SUB1 I) (SUB1 J))) (EQUAL (NTH (SUB1 I) (PUT VAL (SUB1 J) (CDR ARRAY))) (NTH (SUB1 I) (CDR ARRAY))) (NUMBERP I) (NUMBERP J) (LESSP (SUB1 J) (LENGTH (CDR ARRAY))) (EQUAL I J) (NOT (EQUAL I 0))) (EQUAL (NTH (SUB1 I) (CDR ARRAY)) VAL)), which again simplifies, trivially, to: T. Case 1.3. (IMPLIES (AND (LISTP ARRAY) (NOT (EQUAL (ADD1 (LENGTH (CDR ARRAY))) 0)) (NOT (EQUAL J 0)) (NOT (EQUAL (SUB1 I) (SUB1 J))) (EQUAL (NTH (SUB1 I) (PUT VAL (SUB1 J) (CDR ARRAY))) (NTH (SUB1 I) (CDR ARRAY))) (NUMBERP I) (NUMBERP J) (LESSP (SUB1 J) (LENGTH (CDR ARRAY))) (EQUAL I J) (EQUAL I 0)) (EQUAL (CAR ARRAY) VAL)). This again simplifies, obviously, to: T. Case 1.2. (IMPLIES (AND (LISTP ARRAY) (NOT (EQUAL (ADD1 (LENGTH (CDR ARRAY))) 0)) (NOT (EQUAL J 0)) (EQUAL (SUB1 I) (SUB1 J)) (EQUAL (NTH (SUB1 I) (PUT VAL (SUB1 J) (CDR ARRAY))) VAL) (NUMBERP I) (NUMBERP J) (LESSP (SUB1 I) (LENGTH (CDR ARRAY))) (NOT (EQUAL I J)) (NOT (EQUAL I 0))) (EQUAL VAL (NTH (SUB1 I) (CDR ARRAY)))). This again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (LISTP ARRAY) (NOT (EQUAL (ADD1 (LENGTH (CDR ARRAY))) 0)) (NOT (EQUAL J 0)) (EQUAL (SUB1 I) (SUB1 J)) (EQUAL (NTH (SUB1 I) (PUT VAL (SUB1 J) (CDR ARRAY))) VAL) (NUMBERP I) (NUMBERP J) (LESSP (SUB1 I) (LENGTH (CDR ARRAY))) (EQUAL I J) (EQUAL I 0)) (EQUAL (CAR ARRAY) VAL)), which again simplifies, clearly, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.1 ] NTH-PUT (DEFN TIC-TAC-TOE1-C (XMOVES BOARD I T3) (LET ((BOARD1 (MOVEN 1 (CAR XMOVES) BOARD))) (IF (LEGALPN (CAR XMOVES) BOARD) (IF (LESSP I 9) (LET ((BOARD2 (MOVEN 2 (PICK-MOVEN BOARD1) BOARD1))) (IF (WINPN 2 BOARD2) (PUT -2 (ADD1 (ADD1 I)) (PUT BOARD2 (ADD1 I) (PUT BOARD1 I T3))) (TIC-TAC-TOE1-C (CDR XMOVES) BOARD2 (ADD1 (ADD1 I)) (PUT BOARD2 (ADD1 I) (PUT BOARD1 I T3))))) (PUT -1 (ADD1 I) (PUT BOARD1 I T3))) (PUT -2 (ADD1 I) (PUT BOARD1 I T3)))) ((LESSP (DIFFERENCE 9 I)))) Linear arithmetic, the lemma SUB1-TYPE-RESTRICTION, and the definitions of LESSP, ADD1, DIFFERENCE, WINPN, NTHN, EQUAL, WINP1N, and LEGALPN inform us that the measure (DIFFERENCE 9 I) decreases according to the well-founded relation LESSP in each recursive call. Hence, TIC-TAC-TOE1-C is accepted under the definitional principle. From the definition we can conclude that: (LISTP (TIC-TAC-TOE1-C XMOVES BOARD I T3)) is a theorem. [ 0.0 0.1 0.0 ] TIC-TAC-TOE1-C (DEFN TIC-TAC-TOE-C (XMOVES) (TIC-TAC-TOE1-C XMOVES 0 1 (LIST 0 0 0 0 0 0 0 0 0 0 0))) Note that (LISTP (TIC-TAC-TOE-C XMOVES)) is a theorem. [ 0.0 0.0 0.0 ] TIC-TAC-TOE-C (PROVE-LEMMA LENGTH-PUT (REWRITE) (IMPLIES (LESSP I (LENGTH ARRAY)) (EQUAL (LENGTH (PUT VAL I ARRAY)) (LENGTH ARRAY)))) Name the conjecture *1. Perhaps we can prove it by induction. Four inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY)))) (p VAL I ARRAY)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p VAL I ARRAY)) (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (p VAL (SUB1 I) (CDR ARRAY))) (p VAL I ARRAY))). Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions of OR and NOT inform us that the measure (COUNT I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for ARRAY. The above induction scheme leads to the following four new conjectures: Case 4. (IMPLIES (AND (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY)))) (LESSP I (LENGTH ARRAY))) (EQUAL (LENGTH (PUT VAL I ARRAY)) (LENGTH ARRAY))). This simplifies, opening up LENGTH, NOT, OR, EQUAL, and LESSP, to: T. Case 3. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (OR (EQUAL I 0) (NOT (NUMBERP I))) (LESSP I (LENGTH ARRAY))) (EQUAL (LENGTH (PUT VAL I ARRAY)) (LENGTH ARRAY))). This simplifies, rewriting with CDR-CONS, and expanding the functions LENGTH, NOT, OR, EQUAL, LESSP, and PUT, to: T. Case 2. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (NOT (LESSP (SUB1 I) (LENGTH (CDR ARRAY)))) (LESSP I (LENGTH ARRAY))) (EQUAL (LENGTH (PUT VAL I ARRAY)) (LENGTH ARRAY))), which simplifies, rewriting with SUB1-ADD1, and unfolding LENGTH, NOT, OR, and LESSP, to: T. Case 1. (IMPLIES (AND (NOT (OR (EQUAL (LENGTH ARRAY) 0) (NOT (NUMBERP (LENGTH ARRAY))))) (NOT (OR (EQUAL I 0) (NOT (NUMBERP I)))) (EQUAL (LENGTH (PUT VAL (SUB1 I) (CDR ARRAY))) (LENGTH (CDR ARRAY))) (LESSP I (LENGTH ARRAY))) (EQUAL (LENGTH (PUT VAL I ARRAY)) (LENGTH ARRAY))). This simplifies, applying SUB1-ADD1 and CDR-CONS, and expanding LENGTH, NOT, OR, LESSP, and PUT, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] LENGTH-PUT (PROVE-LEMMA ARRAY->LST-OPENER (REWRITE) (IMPLIES (AND (LESSP J (LENGTH ARRAY)) (LESSP I (LENGTH ARRAY))) (EQUAL (ARRAY->LST I (PUT VAL J ARRAY)) (IF (NUMBERP (NTH I (PUT VAL J ARRAY))) (CONS (NTH I (PUT VAL J ARRAY)) (ARRAY->LST (ADD1 I) (PUT VAL J ARRAY))) (LIST (NTH I (PUT VAL J ARRAY))))))) This formula simplifies, rewriting with the lemma LENGTH-PUT, and opening up the function ARRAY->LST, to: T. Q.E.D. [ 0.0 0.1 0.0 ] ARRAY->LST-OPENER (PROVE-LEMMA ARRAY->LST-OPENER-STOPPER (REWRITE) (IMPLIES (AND (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY)) (LESSP J I)) (EQUAL (ARRAY->LST I (PUT VAL J ARRAY)) (ARRAY->LST I ARRAY)))) Give the conjecture the name *1. We will appeal to induction. There are seven plausible inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (NOT (LESSP I (LENGTH (PUT VAL J ARRAY)))) (p I VAL J ARRAY)) (IMPLIES (AND (LESSP I (LENGTH (PUT VAL J ARRAY))) (NUMBERP (NTH I (PUT VAL J ARRAY))) (p (ADD1 I) VAL J ARRAY)) (p I VAL J ARRAY)) (IMPLIES (AND (LESSP I (LENGTH (PUT VAL J ARRAY))) (NOT (NUMBERP (NTH I (PUT VAL J ARRAY))))) (p I VAL J ARRAY))). Linear arithmetic informs us that the measure: (DIFFERENCE (LENGTH (PUT VAL J ARRAY)) I) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction scheme generates the following four new formulas: Case 4. (IMPLIES (AND (NOT (LESSP I (LENGTH (PUT VAL J ARRAY)))) (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY)) (LESSP J I)) (EQUAL (ARRAY->LST I (PUT VAL J ARRAY)) (ARRAY->LST I ARRAY))). This simplifies, rewriting with LENGTH-PUT, and unfolding the definitions of ARRAY->LST and EQUAL, to: T. Case 3. (IMPLIES (AND (LESSP I (LENGTH (PUT VAL J ARRAY))) (NUMBERP (NTH I (PUT VAL J ARRAY))) (NOT (LESSP J (ADD1 I))) (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY)) (LESSP J I)) (EQUAL (ARRAY->LST I (PUT VAL J ARRAY)) (ARRAY->LST I ARRAY))), which simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP I (LENGTH (PUT VAL J ARRAY))) (NUMBERP (NTH I (PUT VAL J ARRAY))) (EQUAL (ARRAY->LST (ADD1 I) (PUT VAL J ARRAY)) (ARRAY->LST (ADD1 I) ARRAY)) (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY)) (LESSP J I)) (EQUAL (ARRAY->LST I (PUT VAL J ARRAY)) (ARRAY->LST I ARRAY))), which simplifies, applying LENGTH-PUT, NTH-PUT, ARRAY->LST-OPENER, and CAR-CONS, and opening up the function ARRAY->LST, to the following two new conjectures: Case 2.2. (IMPLIES (AND (LESSP I (LENGTH ARRAY)) (EQUAL I J) (NUMBERP VAL) (EQUAL (ARRAY->LST (ADD1 I) (PUT VAL J ARRAY)) (ARRAY->LST (ADD1 I) ARRAY)) (NUMBERP I) (LESSP I I) (NOT (NUMBERP (NTH I ARRAY)))) (EQUAL (CONS VAL (ARRAY->LST (ADD1 I) (PUT VAL I ARRAY))) (LIST (NTH I ARRAY)))). However this again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (LESSP I (LENGTH ARRAY)) (EQUAL I J) (NUMBERP VAL) (EQUAL (ARRAY->LST (ADD1 I) (PUT VAL J ARRAY)) (ARRAY->LST (ADD1 I) ARRAY)) (NUMBERP I) (LESSP I I) (NUMBERP (NTH I ARRAY))) (EQUAL (CONS VAL (ARRAY->LST (ADD1 I) (PUT VAL I ARRAY))) (CONS (NTH I ARRAY) (ARRAY->LST (ADD1 I) ARRAY)))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP I (LENGTH (PUT VAL J ARRAY))) (NOT (NUMBERP (NTH I (PUT VAL J ARRAY)))) (NUMBERP I) (NUMBERP J) (LESSP J (LENGTH ARRAY)) (LESSP J I)) (EQUAL (ARRAY->LST I (PUT VAL J ARRAY)) (ARRAY->LST I ARRAY))), which simplifies, applying LENGTH-PUT, NTH-PUT, ARRAY->LST-OPENER, and CAR-CONS, and opening up ARRAY->LST, to the following two new goals: Case 1.2. (IMPLIES (AND (LESSP I (LENGTH ARRAY)) (EQUAL I J) (NOT (NUMBERP VAL)) (NUMBERP I) (LESSP I I) (NOT (NUMBERP (NTH I ARRAY)))) (EQUAL (LIST VAL) (LIST (NTH I ARRAY)))). This again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (LESSP I (LENGTH ARRAY)) (EQUAL I J) (NOT (NUMBERP VAL)) (NUMBERP I) (LESSP I I) (NUMBERP (NTH I ARRAY))) (EQUAL (LIST VAL) (CONS (NTH I ARRAY) (ARRAY->LST (ADD1 I) ARRAY)))), which again simplifies, using linear arithmetic, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.0 ] ARRAY->LST-OPENER-STOPPER (PROVE-LEMMA LENGTH-TIC-TAC-TOE1-C (REWRITE) (IMPLIES (AND (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)) ((DISABLE LEGALPN WINPN PUT NTH))) . Applying the lemma REMAINDER-QUOTIENT-ELIM, replace I by (PLUS X (TIMES 2 Z)) to eliminate (REMAINDER I 2) and (QUOTIENT I 2). We employ LESSP-REMAINDER, the type restriction lemma noted when REMAINDER was introduced, and the type restriction lemma noted when QUOTIENT was introduced to restrict the new variables. We thus obtain: (IMPLIES (AND (NUMBERP X) (EQUAL (LESSP X 2) (NOT (ZEROP 2))) (NUMBERP Z) (NUMBERP BOARD) (EQUAL X 1) (LESSP (PLUS X (TIMES 2 Z)) 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD (PLUS X (TIMES 2 Z)) T3)) 11)), which simplifies, opening up the definitions of NUMBERP, LESSP, ZEROP, NOT, and EQUAL, to the conjecture: (IMPLIES (AND (NUMBERP Z) (NUMBERP BOARD) (LESSP (PLUS 1 (TIMES 2 Z)) 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD (PLUS 1 (TIMES 2 Z)) T3)) 11)), which we would usually push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us consider: (IMPLIES (AND (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)), which we named *1 above. We will appeal to induction. There are four plausible inductions, all of which are unflawed. We limit our consideration to the two suggested by the largest number of nonprimitive recursive functions in the conjecture. However, one of these is more likely than the other. We will induct according to the following scheme: (AND (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (p XMOVES BOARD I T3)) (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (p (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) (p XMOVES BOARD I T3)) (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NOT (LESSP I 9))) (p XMOVES BOARD I T3)) (IMPLIES (NOT (LEGALPN (CAR XMOVES) BOARD)) (p XMOVES BOARD I T3))). Linear arithmetic, the lemma SUB1-TYPE-RESTRICTION, and the definitions of LESSP, ADD1, DIFFERENCE, WINPN, NTHN, EQUAL, WINP1N, and LEGALPN can be used to prove that the measure (DIFFERENCE 9 I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instances chosen for XMOVES, BOARD, and T3. The above induction scheme produces seven new formulas: Case 7. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)), which simplifies, using linear arithmetic, applying LENGTH-PUT, and expanding TIC-TAC-TOE1-C and EQUAL, to: T. Case 6. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (EQUAL (REMAINDER (ADD1 (ADD1 I)) 2) 1)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)). This simplifies, rewriting with the lemma SUB1-ADD1, and expanding the definitions of DIFFERENCE, LESSP, SUB1, NUMBERP, EQUAL, and REMAINDER, to: T. Case 5. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (LESSP (ADD1 (ADD1 I)) 10)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)). This simplifies, using linear arithmetic, to: (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP 8 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (LESSP 10 10)) (NUMBERP BOARD) (NUMBERP 8) (EQUAL (REMAINDER 8 2) 1) (LESSP 8 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD 8 T3)) 11)), which again simplifies, unfolding the definitions of LESSP, NUMBERP, REMAINDER, and EQUAL, to: T. Case 4. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (EQUAL (LENGTH (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3))) 11)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)), which simplifies, using linear arithmetic, applying LENGTH-PUT, and expanding EQUAL, to: T. Case 3. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (EQUAL (LENGTH (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) 11) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)). This simplifies, expanding the definitions of TIC-TAC-TOE1-C and EQUAL, to: T. Case 2. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NOT (LESSP I 9)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)). This simplifies, using linear arithmetic, to the new formula: (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NOT (LESSP 9 9)) (NUMBERP BOARD) (NUMBERP 9) (EQUAL (REMAINDER 9 2) 1) (LESSP 9 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD 9 T3)) 11)), which again simplifies, appealing to the lemma LENGTH-PUT, and expanding the definitions of LESSP, NUMBERP, REMAINDER, EQUAL, ADD1, and TIC-TAC-TOE1-C, to: T. Case 1. (IMPLIES (AND (NOT (LEGALPN (CAR XMOVES) BOARD)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (LENGTH (TIC-TAC-TOE1-C XMOVES BOARD I T3)) 11)), which simplifies, using linear arithmetic, applying LENGTH-PUT, and expanding the definitions of TIC-TAC-TOE1-C and EQUAL, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.4 0.0 ] LENGTH-TIC-TAC-TOE1-C (PROVE-LEMMA NTH-TIC-TAC-TOE1-C (REWRITE) (IMPLIES (AND (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I)) (EQUAL (NTH N (TIC-TAC-TOE1-C XMOVES BOARD I T3)) (NTH N T3))) ((INDUCT (TIC-TAC-TOE1-C XMOVES BOARD I T3)) (DISABLE LEGALPN WINPN PUT NTH))) This conjecture can be simplified, using the abbreviations IMPLIES, NOT, OR, and AND, to four new formulas: Case 4. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I)) (EQUAL (NTH N (TIC-TAC-TOE1-C XMOVES BOARD I T3)) (NTH N T3))), which simplifies, using linear arithmetic, applying LENGTH-PUT and NTH-PUT, and opening up TIC-TAC-TOE1-C, to the following three new conjectures: Case 4.3. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I) (NOT (EQUAL N (ADD1 (ADD1 I)))) (EQUAL N (ADD1 I))) (EQUAL (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (NTH N T3))). This again simplifies, using linear arithmetic, to: T. Case 4.2. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I) (NOT (EQUAL N (ADD1 (ADD1 I)))) (NOT (EQUAL N (ADD1 I))) (EQUAL N I)) (EQUAL (MOVEN 1 (CAR XMOVES) BOARD) (NTH N T3))), which again simplifies, using linear arithmetic, to: T. Case 4.1. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I) (EQUAL N (ADD1 (ADD1 I)))) (EQUAL -2 (NTH N T3))), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (IMPLIES (AND (NUMBERP (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD))) (NUMBERP (ADD1 (ADD1 I))) (EQUAL (REMAINDER (ADD1 (ADD1 I)) 2) 1) (LESSP (ADD1 (ADD1 I)) 10) (EQUAL (LENGTH (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3))) 11) (NUMBERP N) (LESSP N (ADD1 (ADD1 I)))) (EQUAL (NTH N (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) (NTH N (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3))))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I)) (EQUAL (NTH N (TIC-TAC-TOE1-C XMOVES BOARD I T3)) (NTH N T3))), which simplifies, using linear arithmetic, applying SUB1-ADD1, LENGTH-PUT, and NTH-PUT, and unfolding the functions DIFFERENCE, LESSP, SUB1, NUMBERP, EQUAL, REMAINDER, AND, IMPLIES, and TIC-TAC-TOE1-C, to the following four new conjectures: Case 3.4. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (LESSP I 8)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I)) (EQUAL (NTH N (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) (NTH N T3))). But this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP 8 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (LESSP 8 8)) (NUMBERP BOARD) (NUMBERP 8) (EQUAL (REMAINDER 8 2) 1) (LESSP 8 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N 8)) (EQUAL (NTH N (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) 10 (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) 9 (PUT (MOVEN 1 (CAR XMOVES) BOARD) 8 T3)))) (NTH N T3))). But this again simplifies, expanding LESSP, NUMBERP, REMAINDER, and EQUAL, to: T. Case 3.3. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (EQUAL N 0)) (NOT (EQUAL (SUB1 N) 0)) (NOT (LESSP (SUB1 (SUB1 N)) I)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I)) (EQUAL (NTH N (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) (NTH N T3))), which again simplifies, using linear arithmetic, to: T. Case 3.2. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (EQUAL N (ADD1 I))) (EQUAL N I) (EQUAL (NTH N (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) (MOVEN 1 (CAR XMOVES) BOARD)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (LESSP I I)) (EQUAL (NTH I (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) (NTH I T3))), which again simplifies, using linear arithmetic, to: T. Case 3.1. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (EQUAL N (ADD1 I)) (EQUAL (NTH N (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (LESSP N I)) (EQUAL (NTH N (TIC-TAC-TOE1-C XMOVES BOARD I T3)) (NTH N T3))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NOT (LESSP I 9)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I)) (EQUAL (NTH N (TIC-TAC-TOE1-C XMOVES BOARD I T3)) (NTH N T3))), which simplifies, using linear arithmetic, to the formula: (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NOT (LESSP 9 9)) (NUMBERP BOARD) (NUMBERP 9) (EQUAL (REMAINDER 9 2) 1) (LESSP 9 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N 9)) (EQUAL (NTH N (TIC-TAC-TOE1-C XMOVES BOARD 9 T3)) (NTH N T3))). But this again simplifies, applying LENGTH-PUT and NTH-PUT, and expanding the definitions of LESSP, NUMBERP, REMAINDER, EQUAL, ADD1, and TIC-TAC-TOE1-C, to the following two new goals: Case 2.2. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NUMBERP BOARD) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N 9) (NOT (EQUAL N 10)) (EQUAL N 9)) (EQUAL (MOVEN 1 (CAR XMOVES) BOARD) (NTH N T3))). But this again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NUMBERP BOARD) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N 9) (EQUAL N 10)) (EQUAL -1 (NTH N T3))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (LEGALPN (CAR XMOVES) BOARD)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I)) (EQUAL (NTH N (TIC-TAC-TOE1-C XMOVES BOARD I T3)) (NTH N T3))), which simplifies, using linear arithmetic, applying LENGTH-PUT and NTH-PUT, and opening up the definition of TIC-TAC-TOE1-C, to the following two new conjectures: Case 1.2. (IMPLIES (AND (NOT (LEGALPN (CAR XMOVES) BOARD)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I) (NOT (EQUAL N (ADD1 I))) (EQUAL N I)) (EQUAL (MOVEN 1 (CAR XMOVES) BOARD) (NTH N T3))). However this again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (NOT (LEGALPN (CAR XMOVES) BOARD)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11) (NUMBERP N) (LESSP N I) (EQUAL N (ADD1 I))) (EQUAL -2 (NTH N T3))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.1 1.6 0.1 ] NTH-TIC-TAC-TOE1-C (PROVE-LEMMA REMAINDER-ADD1-ADD1-2 (REWRITE) (EQUAL (REMAINDER (ADD1 (ADD1 I)) 2) (REMAINDER I 2))) This formula simplifies, appealing to the lemma SUB1-ADD1, and expanding the functions DIFFERENCE, LESSP, SUB1, NUMBERP, EQUAL, and REMAINDER, to the following two new formulas: Case 2. (IMPLIES (EQUAL I 0) (EQUAL (REMAINDER 0 2) (REMAINDER I 2))). This again simplifies, obviously, to: T. Case 1. (IMPLIES (NOT (NUMBERP I)) (EQUAL (REMAINDER 0 2) (REMAINDER I 2))). However this again simplifies, expanding REMAINDER, LESSP, NUMBERP, and EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] REMAINDER-ADD1-ADD1-2 (PROVE-LEMMA ODD-LESSP-9-IS-LESSP-8 (REWRITE) (IMPLIES (AND (EQUAL (REMAINDER I 2) 1) (LESSP I 9)) (EQUAL (LESSP I 8) T))) This formula simplifies, using linear arithmetic, to the following two new goals: Case 2. (IMPLIES (AND (NOT (NUMBERP I)) (EQUAL (REMAINDER I 2) 1) (LESSP I 9)) (LESSP I 8)). This again simplifies, unfolding the definitions of LESSP, NUMBERP, EQUAL, and REMAINDER, to: T. Case 1. (IMPLIES (AND (NUMBERP I) (EQUAL (REMAINDER 8 2) 1) (LESSP 8 9)) (LESSP 8 8)), which again simplifies, unfolding the definitions of REMAINDER and EQUAL, to: T. Q.E.D. [ 0.0 0.0 0.0 ] ODD-LESSP-9-IS-LESSP-8 (PROVE-LEMMA LEMMA-C1 NIL (IMPLIES (AND (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))) ((DISABLE LEGALPN WINPN PUT NTH))) . Appealing to the lemma REMAINDER-QUOTIENT-ELIM, we now replace I by (PLUS X (TIMES 2 Z)) to eliminate (REMAINDER I 2) and (QUOTIENT I 2). We employ LESSP-REMAINDER, the type restriction lemma noted when REMAINDER was introduced, and the type restriction lemma noted when QUOTIENT was introduced to constrain the new variables. The result is the formula: (IMPLIES (AND (NUMBERP X) (EQUAL (LESSP X 2) (NOT (ZEROP 2))) (NUMBERP Z) (NUMBERP BOARD) (EQUAL X 1) (LESSP (PLUS X (TIMES 2 Z)) 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD (PLUS X (TIMES 2 Z))) (ARRAY->LST (PLUS X (TIMES 2 Z)) (TIC-TAC-TOE1-C XMOVES BOARD (PLUS X (TIMES 2 Z)) T3)))). This simplifies, opening up the functions NUMBERP, LESSP, ZEROP, NOT, and EQUAL, to: (IMPLIES (AND (NUMBERP Z) (NUMBERP BOARD) (LESSP (PLUS 1 (TIMES 2 Z)) 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD (PLUS 1 (TIMES 2 Z))) (ARRAY->LST (PLUS 1 (TIMES 2 Z)) (TIC-TAC-TOE1-C XMOVES BOARD (PLUS 1 (TIMES 2 Z)) T3)))), which we would usually push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us consider: (IMPLIES (AND (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))). We gave this the name *1 above. Perhaps we can prove it by induction. There are five plausible inductions. They merge into four likely candidate inductions, all of which are unflawed. So we will choose the one suggested by the largest number of nonprimitive recursive functions. We will induct according to the following scheme: (AND (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (p XMOVES BOARD I T3)) (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (p (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3)))) (p XMOVES BOARD I T3)) (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NOT (LESSP I 9))) (p XMOVES BOARD I T3)) (IMPLIES (NOT (LEGALPN (CAR XMOVES) BOARD)) (p XMOVES BOARD I T3))). Linear arithmetic, the lemma SUB1-TYPE-RESTRICTION, and the definitions of LESSP, ADD1, DIFFERENCE, WINPN, NTHN, EQUAL, WINP1N, and LEGALPN establish that the measure (DIFFERENCE 9 I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instances chosen for XMOVES, BOARD, and T3. The above induction scheme produces the following seven new formulas: Case 7. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))). This simplifies, using linear arithmetic, applying the lemmas LENGTH-PUT, NTH-PUT, ADD1-EQUAL, and ARRAY->LST-OPENER, and unfolding the functions TIC-TAC-TOE1-B, TIC-TAC-TOE1-C, NUMBERP, and CONS, to: T. Case 6. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (EQUAL (REMAINDER (ADD1 (ADD1 I)) 2) 1)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))). This simplifies, rewriting with REMAINDER-ADD1-ADD1-2, and expanding the definition of EQUAL, to: T. Case 5. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (LESSP (ADD1 (ADD1 I)) 10)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))), which simplifies, using linear arithmetic, to: (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP 8 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (LESSP 10 10)) (NUMBERP BOARD) (NUMBERP 8) (EQUAL (REMAINDER 8 2) 1) (LESSP 8 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD 8) (ARRAY->LST 8 (TIC-TAC-TOE1-C XMOVES BOARD 8 T3)))). But this again simplifies, unfolding the functions LESSP, NUMBERP, REMAINDER, and EQUAL, to: T. Case 4. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (NOT (EQUAL (LENGTH (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3))) 11)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))), which simplifies, using linear arithmetic, applying LENGTH-PUT, and unfolding the function EQUAL, to: T. Case 3. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1-B (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I))) (ARRAY->LST (ADD1 (ADD1 I)) (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3))))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))). This simplifies, using linear arithmetic, rewriting with the lemmas NTH-TIC-TAC-TOE1-C, NTH-PUT, LENGTH-TIC-TAC-TOE1-C, REMAINDER-ADD1-ADD1-2, SUB1-ADD1, ODD-LESSP-9-IS-LESSP-8, and LENGTH-PUT, and expanding the functions TIC-TAC-TOE1-B, TIC-TAC-TOE1-C, EQUAL, SUB1, NUMBERP, LESSP, and ARRAY->LST, to the new conjecture: (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1-B (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I))) (ARRAY->LST (ADD1 (ADD1 I)) (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR XMOVES) BOARD)) (MOVEN 1 (CAR XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR XMOVES) BOARD) I T3))))) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (LESSP I 11)), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NOT (LESSP I 9)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))), which simplifies, using linear arithmetic, to the formula: (IMPLIES (AND (LEGALPN (CAR XMOVES) BOARD) (NOT (LESSP 9 9)) (NUMBERP BOARD) (NUMBERP 9) (EQUAL (REMAINDER 9 2) 1) (LESSP 9 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD 9) (ARRAY->LST 9 (TIC-TAC-TOE1-C XMOVES BOARD 9 T3)))). This again simplifies, appealing to the lemmas LENGTH-PUT, NTH-PUT, and ARRAY->LST-OPENER, and unfolding the functions LESSP, NUMBERP, REMAINDER, EQUAL, TIC-TAC-TOE1-B, ADD1, TIC-TAC-TOE1-C, and CONS, to: T. Case 1. (IMPLIES (AND (NOT (LEGALPN (CAR XMOVES) BOARD)) (NUMBERP BOARD) (NUMBERP I) (EQUAL (REMAINDER I 2) 1) (LESSP I 10) (EQUAL (LENGTH T3) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES BOARD I) (ARRAY->LST I (TIC-TAC-TOE1-C XMOVES BOARD I T3)))), which simplifies, using linear arithmetic, appealing to the lemmas LENGTH-PUT, NTH-PUT, and ARRAY->LST-OPENER, and unfolding the definitions of TIC-TAC-TOE1-B, TIC-TAC-TOE1-C, NUMBERP, and CONS, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.8 0.0 ] LEMMA-C1 (PROVE-LEMMA LEMMA-C (REWRITE) (EQUAL (TIC-TAC-TOE-B XMOVES) (ARRAY->LST 0 (TIC-TAC-TOE-C XMOVES))) ((USE (LEMMA-C1 (BOARD 0) (I 1) (T3 (LIST 0 0 0 0 0 0 0 0 0 0 0)))))) WARNING: Note that the rewrite rule LEMMA-C will be stored so as to apply only to terms with the nonrecursive function symbol TIC-TAC-TOE-B. This conjecture can be simplified, using the abbreviations TIC-TAC-TOE-C and TIC-TAC-TOE-B, to: (IMPLIES (IMPLIES (AND (NUMBERP 0) (NUMBERP 1) (EQUAL (REMAINDER 1 2) 1) (LESSP 1 10) (EQUAL (LENGTH '(0 0 0 0 0 0 0 0 0 0 0)) 11)) (EQUAL (TIC-TAC-TOE1-B XMOVES 0 1) (ARRAY->LST 1 (TIC-TAC-TOE1-C XMOVES 0 1 '(0 0 0 0 0 0 0 0 0 0 0))))) (EQUAL (CONS 0 (TIC-TAC-TOE1-B XMOVES 0 1)) (ARRAY->LST 0 (TIC-TAC-TOE1-C XMOVES 0 1 '(0 0 0 0 0 0 0 0 0 0 0))))). This simplifies, rewriting with LENGTH-TIC-TAC-TOE1-C and NTH-TIC-TAC-TOE1-C, and expanding NUMBERP, REMAINDER, EQUAL, LESSP, LENGTH, AND, ADD1, ARRAY->LST, IMPLIES, and NTH, to: T. Q.E.D. [ 0.0 1.3 0.0 ] LEMMA-C (DEFN TIC-TAC-TOE1-D (XMOVES J BOARD I T3) (LET ((BOARD1 (MOVEN 1 (NTH J XMOVES) BOARD))) (IF (LEGALPN (NTH J XMOVES) BOARD) (IF (LESSP I 9) (LET ((BOARD2 (MOVEN 2 (PICK-MOVEN BOARD1) BOARD1))) (IF (WINPN 2 BOARD2) (PUT -2 (ADD1 (ADD1 I)) (PUT BOARD2 (ADD1 I) (PUT BOARD1 I T3))) (TIC-TAC-TOE1-D XMOVES (ADD1 J) BOARD2 (ADD1 (ADD1 I)) (PUT BOARD2 (ADD1 I) (PUT BOARD1 I T3))))) (PUT -1 (ADD1 I) (PUT BOARD1 I T3))) (PUT -2 (ADD1 I) (PUT BOARD1 I T3)))) ((LESSP (DIFFERENCE 9 I)))) Linear arithmetic, the lemma SUB1-TYPE-RESTRICTION, and the definitions of LESSP, ADD1, DIFFERENCE, WINPN, NTHN, EQUAL, WINP1N, and LEGALPN can be used to show that the measure (DIFFERENCE 9 I) decreases according to the well-founded relation LESSP in each recursive call. Hence, TIC-TAC-TOE1-D is accepted under the definitional principle. From the definition we can conclude that (LISTP (TIC-TAC-TOE1-D XMOVES J BOARD I T3)) is a theorem. [ 0.0 0.1 0.1 ] TIC-TAC-TOE1-D (DEFN TIC-TAC-TOE-D (XMOVES) (TIC-TAC-TOE1-D XMOVES 0 0 1 (LIST 0 0 0 0 0 0 0 0 0 0 0))) Note that (LISTP (TIC-TAC-TOE-D XMOVES)) is a theorem. [ 0.0 0.0 0.0 ] TIC-TAC-TOE-D (DEFN NTH-CDR (J LST) (IF (ZEROP J) LST (NTH-CDR (SUB1 J) (CDR LST)))) Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform us that the measure (COUNT J) decreases according to the well-founded relation LESSP in each recursive call. Hence, NTH-CDR is accepted under the principle of definition. [ 0.0 0.0 0.0 ] NTH-CDR (PROVE-LEMMA CAR-NTH-CDR (REWRITE) (EQUAL (CAR (NTH-CDR J LST)) (NTH J LST))) Name the conjecture *1. Perhaps we can prove it by induction. There are two plausible inductions. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP J) (p J LST)) (IMPLIES (AND (NOT (ZEROP J)) (p (SUB1 J) (CDR LST))) (p J LST))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP establish that the measure (COUNT J) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for LST. The above induction scheme produces the following two new goals: Case 2. (IMPLIES (ZEROP J) (EQUAL (CAR (NTH-CDR J LST)) (NTH J LST))). This simplifies, expanding ZEROP, EQUAL, NTH-CDR, and NTH, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP J)) (EQUAL (CAR (NTH-CDR (SUB1 J) (CDR LST))) (NTH (SUB1 J) (CDR LST)))) (EQUAL (CAR (NTH-CDR J LST)) (NTH J LST))). This simplifies, unfolding ZEROP, NTH-CDR, and NTH, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] CAR-NTH-CDR (PROVE-LEMMA CDR-NTH-CDR (REWRITE) (EQUAL (CDR (NTH-CDR J LST)) (NTH-CDR J (CDR LST)))) . Applying the lemma CAR-CDR-ELIM, replace LST by (CONS Z X) to eliminate (CDR LST) and (CAR LST). This produces the following two new goals: Case 2. (IMPLIES (NOT (LISTP LST)) (EQUAL (CDR (NTH-CDR J LST)) (NTH-CDR J (CDR LST)))). However this simplifies, applying CDR-NLISTP, to: (IMPLIES (NOT (LISTP LST)) (EQUAL (CDR (NTH-CDR J LST)) (NTH-CDR J 0))), which we would normally push and work on later by induction. But if we must use induction to prove the input conjecture, we prefer to induct on the original formulation of the problem. Thus we will disregard all that we have previously done, give the name *1 to the original input, and work on it. So now let us return to: (EQUAL (CDR (NTH-CDR J LST)) (NTH-CDR J (CDR LST))), named *1. Let us appeal to the induction principle. Two inductions are suggested by terms in the conjecture. However, they merge into one likely candidate induction. We will induct according to the following scheme: (AND (IMPLIES (ZEROP J) (p J LST)) (IMPLIES (AND (NOT (ZEROP J)) (p (SUB1 J) (CDR LST))) (p J LST))). Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform us that the measure (COUNT J) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instance chosen for LST. The above induction scheme produces two new formulas: Case 2. (IMPLIES (ZEROP J) (EQUAL (CDR (NTH-CDR J LST)) (NTH-CDR J (CDR LST)))), which simplifies, opening up the definitions of ZEROP, EQUAL, and NTH-CDR, to: T. Case 1. (IMPLIES (AND (NOT (ZEROP J)) (EQUAL (CDR (NTH-CDR (SUB1 J) (CDR LST))) (NTH-CDR (SUB1 J) (CDDR LST)))) (EQUAL (CDR (NTH-CDR J LST)) (NTH-CDR J (CDR LST)))), which simplifies, unfolding the functions ZEROP and NTH-CDR, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.1 ] CDR-NTH-CDR (PROVE-LEMMA LEMMA-D1 NIL (EQUAL (TIC-TAC-TOE1-C (NTH-CDR J XMOVES) BOARD I T3) (TIC-TAC-TOE1-D XMOVES J BOARD I T3)) ((DISABLE LEGALPN WINPN PUT NTH))) Give the conjecture the name *1. We will appeal to induction. There are three plausible inductions. They merge into two likely candidate inductions. However, only one is unflawed. We will induct according to the following scheme: (AND (IMPLIES (AND (LEGALPN (CAR (NTH-CDR J XMOVES)) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)))) (p J XMOVES BOARD I T3)) (IMPLIES (AND (LEGALPN (CAR (NTH-CDR J XMOVES)) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)))) (p (ADD1 J) XMOVES (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD) I T3)))) (p J XMOVES BOARD I T3)) (IMPLIES (AND (LEGALPN (CAR (NTH-CDR J XMOVES)) BOARD) (NOT (LESSP I 9))) (p J XMOVES BOARD I T3)) (IMPLIES (NOT (LEGALPN (CAR (NTH-CDR J XMOVES)) BOARD)) (p J XMOVES BOARD I T3))). Linear arithmetic, the lemma SUB1-TYPE-RESTRICTION, and the definitions of LESSP, ADD1, DIFFERENCE, WINPN, NTHN, EQUAL, WINP1N, and LEGALPN establish that the measure (DIFFERENCE 9 I) decreases according to the well-founded relation LESSP in each induction step of the scheme. Note, however, the inductive instances chosen for J, BOARD, and T3. The above induction scheme produces the following four new formulas: Case 4. (IMPLIES (AND (LEGALPN (CAR (NTH-CDR J XMOVES)) BOARD) (LESSP I 9) (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)))) (EQUAL (TIC-TAC-TOE1-C (NTH-CDR J XMOVES) BOARD I T3) (TIC-TAC-TOE1-D XMOVES J BOARD I T3))). This simplifies, rewriting with the lemma CAR-NTH-CDR, and unfolding TIC-TAC-TOE1-C and TIC-TAC-TOE1-D, to: T. Case 3. (IMPLIES (AND (LEGALPN (CAR (NTH-CDR J XMOVES)) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)))) (EQUAL (TIC-TAC-TOE1-C (NTH-CDR (ADD1 J) XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD) I T3))) (TIC-TAC-TOE1-D XMOVES (ADD1 J) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD) I T3))))) (EQUAL (TIC-TAC-TOE1-C (NTH-CDR J XMOVES) BOARD I T3) (TIC-TAC-TOE1-D XMOVES J BOARD I T3))). This simplifies, rewriting with CAR-NTH-CDR and CDR-NTH-CDR, and expanding the functions TIC-TAC-TOE1-C and TIC-TAC-TOE1-D, to: (IMPLIES (AND (LEGALPN (NTH J XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)))) (EQUAL (TIC-TAC-TOE1-C (NTH-CDR (ADD1 J) XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD) I T3))) (TIC-TAC-TOE1-D XMOVES (ADD1 J) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD)) (ADD1 I) (PUT (MOVEN 1 (CAR (NTH-CDR J XMOVES)) BOARD) I T3))))) (EQUAL (TIC-TAC-TOE1-C (NTH-CDR J (CDR XMOVES)) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (NTH J XMOVES) BOARD) I T3))) (TIC-TAC-TOE1-D XMOVES (ADD1 J) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (NTH J XMOVES) BOARD) I T3))))). However this further simplifies, applying SUB1-ADD1, CAR-NTH-CDR, and SUB1-TYPE-RESTRICTION, and unfolding the function NTH-CDR, to the new goal: (IMPLIES (AND (LEGALPN (NTH J XMOVES) BOARD) (LESSP I 9) (NOT (WINPN 2 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)))) (NOT (NUMBERP J)) (EQUAL (TIC-TAC-TOE1-C (NTH-CDR 0 (CDR XMOVES)) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (NTH J XMOVES) BOARD) I T3))) (TIC-TAC-TOE1-D XMOVES (ADD1 J) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (NTH J XMOVES) BOARD) I T3))))) (EQUAL (TIC-TAC-TOE1-C (CDR XMOVES) (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (NTH J XMOVES) BOARD) I T3))) (TIC-TAC-TOE1-D XMOVES 1 (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 (ADD1 I)) (PUT (MOVEN 2 (PICK-MOVEN (MOVEN 1 (NTH J XMOVES) BOARD)) (MOVEN 1 (NTH J XMOVES) BOARD)) (ADD1 I) (PUT (MOVEN 1 (NTH J XMOVES) BOARD) I T3))))), which again simplifies, appealing to the lemma SUB1-TYPE-RESTRICTION, and opening up EQUAL and NTH-CDR, to: T. Case 2. (IMPLIES (AND (LEGALPN (CAR (NTH-CDR J XMOVES)) BOARD) (NOT (LESSP I 9))) (EQUAL (TIC-TAC-TOE1-C (NTH-CDR J XMOVES) BOARD I T3) (TIC-TAC-TOE1-D XMOVES J BOARD I T3))), which simplifies, appealing to the lemma CAR-NTH-CDR, and opening up TIC-TAC-TOE1-C and TIC-TAC-TOE1-D, to: T. Case 1. (IMPLIES (NOT (LEGALPN (CAR (NTH-CDR J XMOVES)) BOARD)) (EQUAL (TIC-TAC-TOE1-C (NTH-CDR J XMOVES) BOARD I T3) (TIC-TAC-TOE1-D XMOVES J BOARD I T3))), which simplifies, applying CAR-NTH-CDR, and unfolding the functions TIC-TAC-TOE1-C and TIC-TAC-TOE1-D, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 2.2 0.0 ] LEMMA-D1 (PROVE-LEMMA LEMMA-D (REWRITE) (EQUAL (TIC-TAC-TOE-C XMOVES) (TIC-TAC-TOE-D XMOVES)) ((USE (LEMMA-D1 (J 0) (BOARD 0) (I 1) (T3 (LIST 0 0 0 0 0 0 0 0 0 0 0)))))) WARNING: Note that the rewrite rule LEMMA-D will be stored so as to apply only to terms with the nonrecursive function symbol TIC-TAC-TOE-C. This conjecture can be simplified, using the abbreviations TIC-TAC-TOE-D and TIC-TAC-TOE-C, to: (IMPLIES (EQUAL (TIC-TAC-TOE1-C (NTH-CDR 0 XMOVES) 0 1 '(0 0 0 0 0 0 0 0 0 0 0)) (TIC-TAC-TOE1-D XMOVES 0 0 1 '(0 0 0 0 0 0 0 0 0 0 0))) (EQUAL (TIC-TAC-TOE1-C XMOVES 0 1 '(0 0 0 0 0 0 0 0 0 0 0)) (TIC-TAC-TOE1-D XMOVES 0 0 1 '(0 0 0 0 0 0 0 0 0 0 0)))). This simplifies, opening up the functions EQUAL and NTH-CDR, to: T. Q.E.D. [ 0.0 0.2 0.0 ] LEMMA-D (DEFN UP (T3) (NUMBER->BOARD* (ARRAY->LST 0 T3) 9)) Observe that (OR (LITATOM (UP T3)) (LISTP (UP T3))) is a theorem. [ 0.0 0.0 0.0 ] UP (PROVE-LEMMA TIC-TAC-TOE-D-CORRECT NIL (AND (MEMBER (TIC-TAC-TOEP (UP (TIC-TAC-TOE-D XMOVES))) '(DRAW O-WIN)) (X-MOVES-HONORED XMOVES (UP (TIC-TAC-TOE-D XMOVES))) (EQUAL (INIT-GAME (LENGTH OPENING) (UP (TIC-TAC-TOE-D (APPEND OPENING XMOVES1)))) (INIT-GAME (LENGTH OPENING) (UP (TIC-TAC-TOE-D (APPEND OPENING XMOVES2)))))) ((DISABLE TIC-TAC-TOE TIC-TAC-TOE-A TIC-TAC-TOE-B TIC-TAC-TOE-C TIC-TAC-TOE-D) (USE (TIC-TAC-TOE-CORRECT)))) This conjecture can be simplified, using the abbreviations AND, UP, LEMMA-D, LEMMA-C, LEMMA-B, and LEMMA-A, to: (IMPLIES (AND (MEMBER (TIC-TAC-TOEP (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D XMOVES)) 9)) '(DRAW O-WIN)) (X-MOVES-HONORED XMOVES (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D XMOVES)) 9)) (EQUAL (INIT-GAME (LENGTH OPENING) (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES1))) 9)) (INIT-GAME (LENGTH OPENING) (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES2))) 9)))) (AND (MEMBER (TIC-TAC-TOEP (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D XMOVES)) 9)) '(DRAW O-WIN)) (X-MOVES-HONORED XMOVES (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D XMOVES)) 9)) (EQUAL (INIT-GAME (LENGTH OPENING) (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES1))) 9)) (INIT-GAME (LENGTH OPENING) (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES2))) 9))))). This simplifies, rewriting with the lemmas CAR-CONS, CONS-EQUAL, CDR-NLISTP, and CAR-NLISTP, and unfolding the functions TIC-TAC-TOEP, X-MOVES-HONORED, INIT-GAME, MEMBER, AND, X-MOVES-HONORED1, LISTP, and EQUAL, to the following two new goals: Case 2. (IMPLIES (AND (NOT (EQUAL (CAR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D XMOVES)) 9)) (LIST F F F F F F F F F))) (MEMBER F '(DRAW O-WIN)) (X-MOVES-HONORED1 (CAR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D XMOVES)) 9)) XMOVES (CDR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D XMOVES)) 9))) (EQUAL (CAR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES1))) 9)) (CAR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES2))) 9)))) (NOT (EQUAL (INIT-GAME1 (LENGTH OPENING) (CDR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES1))) 9))) (INIT-GAME1 (LENGTH OPENING) (CDR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES2))) 9)))))). But this again simplifies, unfolding MEMBER, to: T. Case 1. (IMPLIES (AND (NOT (LISTP (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D XMOVES)) 9))) (MEMBER F '(DRAW O-WIN)) (EQUAL (CAR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES1))) 9)) (CAR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES2))) 9)))) (NOT (EQUAL (INIT-GAME1 (LENGTH OPENING) (CDR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES1))) 9))) (INIT-GAME1 (LENGTH OPENING) (CDR (NUMBER->BOARD* (ARRAY->LST 0 (TIC-TAC-TOE-D (APPEND OPENING XMOVES2))) 9)))))), which again simplifies, expanding the function MEMBER, to: T. Q.E.D. [ 0.0 0.2 0.0 ] TIC-TAC-TOE-D-CORRECT