(BOOT-STRAP NQTHM) [ 0.0 0.1 0.0 ] GROUND-ZERO (DEFN DOUBLE-LIST (L) (IF (NLISTP L) NIL (CONS (TIMES 2 (CAR L)) (DOUBLE-LIST (CDR L))))) Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP inform us that the measure (COUNT L) decreases according to the well-founded relation LESSP in each recursive call. Hence, DOUBLE-LIST is accepted under the principle of definition. Note that: (OR (LITATOM (DOUBLE-LIST L)) (LISTP (DOUBLE-LIST L))) is a theorem. [ 0.0 0.0 0.0 ] DOUBLE-LIST (PROVE-LEMMA DOUBLE-LIST-APPEND NIL (EQUAL (DOUBLE-LIST (APPEND A B)) (APPEND (DOUBLE-LIST A) (DOUBLE-LIST B)))) Call the conjecture *1. Perhaps we can prove it by induction. 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 (AND (LISTP A) (p (CDR A) B)) (p A B)) (IMPLIES (NOT (LISTP A)) (p A B))). Linear arithmetic and the lemma CDR-LESSP can be used to prove that the measure (COUNT A) 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 (LISTP A) (EQUAL (DOUBLE-LIST (APPEND (CDR A) B)) (APPEND (DOUBLE-LIST (CDR A)) (DOUBLE-LIST B)))) (EQUAL (DOUBLE-LIST (APPEND A B)) (APPEND (DOUBLE-LIST A) (DOUBLE-LIST B)))), which simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up the definitions of APPEND and DOUBLE-LIST, to: T. Case 1. (IMPLIES (NOT (LISTP A)) (EQUAL (DOUBLE-LIST (APPEND A B)) (APPEND (DOUBLE-LIST A) (DOUBLE-LIST B)))), which simplifies, unfolding the functions APPEND, DOUBLE-LIST, and LISTP, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.0 0.0 ] DOUBLE-LIST-APPEND (PROVE-LEMMA SUM-DISTRIBUTES-OVER-PLUS NIL (EQUAL (FOR I L COND 'SUM (LIST 'PLUS G H) A) (PLUS (FOR I L COND 'SUM G A) (FOR I L COND 'SUM H A)))) Call the conjecture *1. 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 L) (p I L COND G H A)) (IMPLIES (AND (NOT (NLISTP L)) (EVAL\$ T COND (CONS (CONS I (CAR L)) A)) (p I (CDR L) COND G H A)) (p I L COND G H A)) (IMPLIES (AND (NOT (NLISTP L)) (NOT (EVAL\$ T COND (CONS (CONS I (CAR L)) A))) (p I (CDR L) COND G H A)) (p I L COND G H A))). Linear arithmetic, the lemmas CDR-LESSEQP and CDR-LESSP, and the definition of NLISTP can be used to establish that the measure (COUNT L) 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 (NLISTP L) (EQUAL (FOR I L COND 'SUM (LIST 'PLUS G H) A) (PLUS (FOR I L COND 'SUM G A) (FOR I L COND 'SUM H A)))). This simplifies, unfolding the functions NLISTP, FOR, QUANTIFIER-INITIAL-VALUE, PLUS, and EQUAL, to: T. Case 2. (IMPLIES (AND (NOT (NLISTP L)) (EVAL\$ T COND (CONS (CONS I (CAR L)) A)) (EQUAL (FOR I (CDR L) COND 'SUM (LIST 'PLUS G H) A) (PLUS (FOR I (CDR L) COND 'SUM G A) (FOR I (CDR L) COND 'SUM H A)))) (EQUAL (FOR I L COND 'SUM (LIST 'PLUS G H) A) (PLUS (FOR I L COND 'SUM G A) (FOR I L COND 'SUM H A)))). This simplifies, applying REWRITE-EVAL\$, CAR-CONS, and CDR-CONS, and unfolding the definitions of NLISTP, FOR, EQUAL, and QUANTIFIER-OPERATION, to: (IMPLIES (AND (LISTP L) (EVAL\$ T COND (CONS (CONS I (CAR L)) A)) (EQUAL (FOR I (CDR L) COND 'SUM (LIST 'PLUS G H) A) (PLUS (FOR I (CDR L) COND 'SUM G A) (FOR I (CDR L) COND 'SUM H A)))) (EQUAL (PLUS (PLUS (EVAL\$ T G (CONS (CONS I (CAR L)) A)) (EVAL\$ T H (CONS (CONS I (CAR L)) A))) (FOR I (CDR L) COND 'SUM (LIST 'PLUS G H) A)) (PLUS (PLUS (EVAL\$ T G (CONS (CONS I (CAR L)) A)) (FOR I (CDR L) COND 'SUM G A)) (EVAL\$ T H (CONS (CONS I (CAR L)) A)) (FOR I (CDR L) COND 'SUM H A)))). But this again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (NOT (NLISTP L)) (NOT (EVAL\$ T COND (CONS (CONS I (CAR L)) A))) (EQUAL (FOR I (CDR L) COND 'SUM (LIST 'PLUS G H) A) (PLUS (FOR I (CDR L) COND 'SUM G A) (FOR I (CDR L) COND 'SUM H A)))) (EQUAL (FOR I L COND 'SUM (LIST 'PLUS G H) A) (PLUS (FOR I L COND 'SUM G A) (FOR I L COND 'SUM H A)))), which simplifies, opening up the definitions of NLISTP and FOR, to: T. That finishes the proof of *1. Q.E.D. [ 0.0 0.1 0.0 ] SUM-DISTRIBUTES-OVER-PLUS (PROVE-LEMMA EVAL\$-DISTRIBUTES-OVER-PLUS NIL (EQUAL (EVAL\$ T (LIST 'PLUS X Y) A) (PLUS (EVAL\$ T X A) (EVAL\$ T Y A)))) This formula simplifies, appealing to the lemmas REWRITE-EVAL\$, CAR-CONS, and CDR-CONS, to: T. Q.E.D. [ 0.0 0.0 0.0 ] EVAL\$-DISTRIBUTES-OVER-PLUS (PROVE-LEMMA V&C\$-LIST-DEFN NIL (EQUAL (V&C\$ 'LIST L VA) (IF (NLISTP L) NIL (CONS (V&C\$ T (CAR L) VA) (V&C\$ 'LIST (CDR L) VA))))) This conjecture simplifies, opening up NLISTP, EQUAL, and V&C\$, to: T. Q.E.D. [ 0.0 0.0 0.0 ] V&C\$-LIST-DEFN (PROVE-LEMMA V&C\$-DEFN NIL (EQUAL (V&C\$ T X VA) (IF (LITATOM X) (CONS (CDR (ASSOC X VA)) 0) (IF (NLISTP X) (CONS X 0) (IF (EQUAL (CAR X) 'QUOTE) (CONS (CADR X) 0) (V&C-APPLY\$ (CAR X) (V&C\$ 'LIST (CDR X) VA))))))) This conjecture simplifies, opening up the definitions of NLISTP, EQUAL, and V&C\$, to: T. Q.E.D. [ 0.0 0.0 0.0 ] V&C\$-DEFN (PROVE-LEMMA EQ-ARGS-GIVE-EQ-VALUES--APPLY-VERSION NIL (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2))) (AND (IFF (V&C-APPLY\$ FN ARGS1) (V&C-APPLY\$ FN ARGS2)) (EQUAL (CAR (V&C-APPLY\$ FN ARGS1)) (CAR (V&C-APPLY\$ FN ARGS2))))) ((ENABLE V&C-APPLY\$))) This simplifies, applying SUB1-ADD1, and expanding the functions FIX-COST, PLUS, V&C-APPLY\$, and AND, to 94 new conjectures: Case 94.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (EQUAL FN 'IF)) (SUBRP FN)) (IFF (CONS (APPLY-SUBR FN (STRIP-CARS ARGS1)) (ADD1 (SUM-CDRS ARGS1))) (CONS (APPLY-SUBR FN (STRIP-CARS ARGS1)) (ADD1 (SUM-CDRS ARGS2))))), which again simplifies, expanding the function IFF, to: T. Case 93.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (NOT (CAR ARGS1))) (IFF F F)), which again simplifies, applying CAR-NLISTP, and opening up the functions EQUAL and MEMBER, to: T. Case 92.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (IFF F F)). But this again simplifies, applying CAR-NLISTP, and unfolding the definitions of EQUAL and MEMBER, to: T. Case 91.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (IFF F F)). But this again simplifies, rewriting with CAR-NLISTP, and unfolding the definitions of EQUAL and MEMBER, to: T. Case 90.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (NOT (CAR ARGS1))) (IFF F F)). But this again simplifies, applying CAR-NLISTP, and unfolding the definitions of EQUAL and MEMBER, to: T. Case 89.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (IFF F F)). But this again simplifies, opening up the functions EQUAL and IFF, to: T. Case 88.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (IFF F F)), which again simplifies, opening up the definitions of EQUAL and IFF, to: T. Case 87.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (NOT (CAR ARGS1))) (IFF F F)), which again simplifies, appealing to the lemma CAR-NLISTP, and opening up EQUAL and MEMBER, to: T. Case 86.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (IFF F F)), which again simplifies, unfolding the definitions of EQUAL and IFF, to: T. Case 85.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (IFF F F)), which again simplifies, expanding the definitions of EQUAL and IFF, to: T. Case 84.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (EQUAL FN 'IF)) (NOT (SUBRP FN)) (NOT (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1))))) (IFF F F)), which again simplifies, opening up the function IFF, to: T. Case 83.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (EQUAL FN 'IF)) (NOT (SUBRP FN)) (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1)))) (IFF (CONS (CAR (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1)))) (ADD1 (PLUS (SUM-CDRS ARGS1) (CDR (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1))))))) (CONS (CAR (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1)))) (ADD1 (PLUS (SUM-CDRS ARGS2) (CDR (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1))))))))), which again simplifies, opening up the definition of IFF, to: T. Case 82.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1)))) (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2)))))), which again simplifies, unfolding EQUAL, PLUS, and IFF, to: T. Case 81.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1)))) (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2)))))), which again simplifies, expanding the definitions of EQUAL, PLUS, and IFF, to: T. Case 80.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAR ARGS1))) (IFF F (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2)))))), which again simplifies, rewriting with the lemma CAR-NLISTP, and opening up the definitions of EQUAL and MEMBER, to: T. Case 79.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1) (NOT (CADR ARGS1))) (IFF F (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2)))))), which again simplifies, unfolding the functions EQUAL, PLUS, and IFF, to the conjecture: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1)) (CADR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We must thus prove seven new formulas: Case 79.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1)) (CADR ARGS1)), which further simplifies, rewriting with CAR-NLISTP, and opening up STRIP-CARS, MEMBER, and CAR, to: T. Case 79.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X))) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 79.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X))) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 79.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X))) (CAAR ARGS1)) (CADR ARGS1)). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: T. Case 79.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding the functions STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X X1) to eliminate (CAR V) and (CDR V), and V by (CONS D C) to eliminate (CAR V) and (CDR V). We thus obtain the following five new goals: Case 79.3.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). This finally simplifies, expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 79.3.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NOT (NUMBERP W)) (CAR X)) (CAR V)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, and EQUAL, to: T. Case 79.3.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS D C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS D C)))) (NOT (NUMBERP W)) (CAR X)) D). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 79.3.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NOT (NUMBERP W)) D) (CAR V)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS and CAR, to: T. Case 79.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X X1)))) (NOT (NUMBERP W)) D) X). This finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS and CAR, to: T. Case 79.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). This further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS V Z1) to eliminate (CAR Z) and (CDR Z), and Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z). We thus obtain the following five new conjectures: Case 79.2.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). However this finally simplifies, unfolding STRIP-CARS, to: T. Case 79.2.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NOT (NUMBERP W)) (CAR V)) (CAR Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 79.2.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS C X1)))) D (NOT (NUMBERP W)) (CAR V)) C). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 79.2.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NOT (NUMBERP W)) C) (CAR Z)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 79.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) D (NOT (NUMBERP W)) C) V). However this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and CAR, to: T. Case 79.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), V by (CONS X V1) to eliminate (CAR V) and (CDR V), and V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V). We thus obtain the following five new goals: Case 79.1.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). However this finally simplifies, expanding the function STRIP-CARS, to: T. Case 79.1.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X V)) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X V))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR X)) (CAR V)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 79.1.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X (CONS X1 Z1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR X)) X1). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 79.1.2. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) V)) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) V))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) X1) (CAR V)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 79.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS X V1))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS X V1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) X1) X). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and CAR, to: T. Case 78.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (IFF F (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2)))))). But this again simplifies, opening up the functions EQUAL, PLUS, and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We must thus prove seven new conjectures: Case 78.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, applying CAR-NLISTP, and opening up STRIP-CARS, MEMBER, and CAR, to: T. Case 78.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 78.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 78.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 78.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X1 X) to eliminate (CDR V) and (CAR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS C D) to eliminate (CDR V) and (CAR V), and D by (CONS V X1) to eliminate (CAR D) and (CDR D). This produces the following seven new formulas: Case 78.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this finally simplifies, expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 78.3.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NOT (NUMBERP W)) (NOT (CAR X))) (CADR V)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CONS, and EQUAL, to: T. Case 78.3.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C D))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C D)))) (NOT (NUMBERP W)) (NOT (CAR X))) (CAR D)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 78.3.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS V X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS V X1))))) (NOT (NUMBERP W)) (NOT (CAR X))) V). This finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 78.3.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NOT (NUMBERP W)) (NOT D)) (CADR V)), which further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding the functions STRIP-CARS, CONS, EQUAL, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (NUMBERP W))) 0). This finally simplifies, trivially, to: T. Case 78.3.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 X))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 X)))) (NOT (NUMBERP W)) (NOT D)) (CAR X)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 78.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS V Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS V Z1))))) (NOT (NUMBERP W)) (NOT D)) V). But this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 78.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). But this further simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS Z1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS X1 C) to eliminate (CDR Z) and (CAR Z), and C by (CONS Z Z1) to eliminate (CAR C) and (CDR C). We must thus prove seven new goals: Case 78.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which finally simplifies, unfolding the definition of STRIP-CARS, to: T. Case 78.2.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NOT (NUMBERP W)) (NOT (CAR V))) (CADR Z)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 78.2.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 C)))) D (NOT (NUMBERP W)) (NOT (CAR V))) (CAR C)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 78.2.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS Z Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS Z Z1))))) D (NOT (NUMBERP W)) (NOT (CAR V))) Z). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 78.2.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NOT (NUMBERP W)) (NOT C)) (CADR Z)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 78.2.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) D (NOT (NUMBERP W)) (NOT C)) (CAR V)). But this further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and opening up STRIP-CARS, EQUAL, and MEMBER, to: (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (CAR Z1) (CAR D)) Z1 D (NOT (NUMBERP W))) 0). This finally simplifies, clearly, to: T. Case 78.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS Z V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS Z V1))))) D (NOT (NUMBERP W)) (NOT C)) Z). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 78.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), V by (CONS V1 X) to eliminate (CDR V) and (CAR V), X by (CONS V W1) to eliminate (CAR X) and (CDR X), V by (CONS Z1 X1) to eliminate (CDR V) and (CAR V), and X1 by (CONS V V1) to eliminate (CAR X1) and (CDR X1). This generates seven new conjectures: Case 78.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which finally simplifies, opening up the function STRIP-CARS, to: T. Case 78.1.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X V)) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X V))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT (CAR X))) (CADR V)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 78.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X (CONS Z1 X1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT (CAR X))) (CAR X1)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 78.1.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS V V1)))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS V V1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT (CAR X))) V). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 78.1.3. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) V)) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) V))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1)) (CADR V)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 78.1.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 X))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 X)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1)) (CAR X)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS and EQUAL, to: T. Case 78.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS V W1)))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS V W1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1)) V). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS, CAR, EQUAL, and MEMBER, to: T. Case 77.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1)))) (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2)))))). However this again simplifies, expanding EQUAL, PLUS, and IFF, to: T. Case 76.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1)))) (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2)))))), which again simplifies, unfolding EQUAL, PLUS, and IFF, to: T. Case 75.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1)))) (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2)))))), which again simplifies, expanding EQUAL, PLUS, and IFF, to: T. Case 74.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1)))) (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2)))))), which again simplifies, expanding EQUAL and IFF, to: T. Case 73.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAR ARGS1))) (IFF F (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2)))))), which again simplifies, applying CAR-NLISTP, and unfolding EQUAL and MEMBER, to: T. Case 72.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (IFF F (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2)))))). However this again simplifies, expanding EQUAL and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1)) (CADR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This generates seven new goals: Case 72.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1)) (CADR ARGS1)), which further simplifies, rewriting with CAR-NLISTP, and opening up STRIP-CARS, MEMBER, and CAR, to: T. Case 72.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X)) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 72.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X)) (CAAR ARGS1)) (CADR ARGS1)), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 72.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X)) (CAAR ARGS1)) (CADR ARGS1)), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and MEMBER, to: T. Case 72.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). This further simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X X1) to eliminate (CAR V) and (CDR V), and V by (CONS D C) to eliminate (CAR V) and (CDR V). This produces the following five new goals: Case 72.3.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). But this finally simplifies, unfolding the functions STRIP-CARS and EQUAL, to: T. Case 72.3.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NUMBERP W) (CAR X)) (CAR V)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 72.3.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS D C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS D C)))) (NUMBERP W) (CAR X)) D). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 72.3.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NUMBERP W) D) (CAR V)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS and CAR, to: T. Case 72.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X X1)))) (NUMBERP W) D) X). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS and CAR, to: T. Case 72.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). However this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS V Z1) to eliminate (CAR Z) and (CDR Z), and Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z). This produces the following five new goals: Case 72.2.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). But this finally simplifies, unfolding the definition of STRIP-CARS, to: T. Case 72.2.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NUMBERP W) (CAR V)) (CAR Z)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 72.2.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS C X1)))) D (NUMBERP W) (CAR V)) C), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CAR, and EQUAL, to: T. Case 72.2.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NUMBERP W) C) (CAR Z)). This finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 72.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) D (NUMBERP W) C) V). But this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding STRIP-CARS and CAR, to: T. Case 72.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS and MEMBER, to the new formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), V by (CONS X V1) to eliminate (CAR V) and (CDR V), and V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V). We thus obtain the following five new conjectures: Case 72.1.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). However this finally simplifies, unfolding the definition of STRIP-CARS, to: T. Case 72.1.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X V)) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X V))) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR X)) (CAR V)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 72.1.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X (CONS X1 Z1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR X)) X1). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 72.1.2. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) V)) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) V))) D (NOT (MEMBER F C)) Z (NUMBERP W) X1) (CAR V)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 72.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS X V1))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS X V1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) X1) X). However this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS and CAR, to: T. Case 71.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (IFF F (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2)))))), which again simplifies, opening up EQUAL and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). The result is seven new conjectures: Case 71.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, rewriting with CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 71.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 71.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 71.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: T. Case 71.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and opening up the functions STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X1 X) to eliminate (CDR V) and (CAR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS C D) to eliminate (CDR V) and (CAR V), and D by (CONS V X1) to eliminate (CAR D) and (CDR D). We thus obtain the following seven new goals: Case 71.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). But this finally simplifies, opening up STRIP-CARS and EQUAL, to: T. Case 71.3.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NUMBERP W) (NOT (CAR X))) (CADR V)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CONS, and EQUAL, to: T. Case 71.3.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C D))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C D)))) (NUMBERP W) (NOT (CAR X))) (CAR D)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 71.3.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS V X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS V X1))))) (NUMBERP W) (NOT (CAR X))) V). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 71.3.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NUMBERP W) (NOT D)) (CADR V)). However this further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, CONS, EQUAL, MEMBER, and CAR, to the conjecture: (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NUMBERP W)) 0). This finally simplifies, clearly, to: T. Case 71.3.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 X))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 X)))) (NUMBERP W) (NOT D)) (CAR X)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 71.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS V Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS V Z1))))) (NUMBERP W) (NOT D)) V). But this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 71.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS Z1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS X1 C) to eliminate (CDR Z) and (CAR Z), and C by (CONS Z Z1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following seven new goals: Case 71.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). This finally simplifies, opening up the function STRIP-CARS, to: T. Case 71.2.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NUMBERP W) (NOT (CAR V))) (CADR Z)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 71.2.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 C)))) D (NUMBERP W) (NOT (CAR V))) (CAR C)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 71.2.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS Z Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS Z Z1))))) D (NUMBERP W) (NOT (CAR V))) Z). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 71.2.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NUMBERP W) (NOT C)) (CADR Z)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 71.2.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) D (NUMBERP W) (NOT C)) (CAR V)). This further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS, EQUAL, and MEMBER, to the goal: (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (CAR Z1) (CAR D)) Z1 D (NUMBERP W)) 0). This finally simplifies, clearly, to: T. Case 71.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS Z V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS Z V1))))) D (NUMBERP W) (NOT C)) Z). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, CAR, and EQUAL, to: T. Case 71.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, applying CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), V by (CONS V1 X) to eliminate (CDR V) and (CAR V), X by (CONS V W1) to eliminate (CAR X) and (CDR X), V by (CONS Z1 X1) to eliminate (CDR V) and (CAR V), and X1 by (CONS V V1) to eliminate (CAR X1) and (CDR X1). We would thus like to prove the following seven new conjectures: Case 71.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). This finally simplifies, unfolding STRIP-CARS, to: T. Case 71.1.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X V)) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X V))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT (CAR X))) (CADR V)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 71.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X (CONS Z1 X1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT (CAR X))) (CAR X1)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 71.1.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS V V1)))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS V V1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT (CAR X))) V), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 71.1.3. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) V)) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) V))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1)) (CADR V)), which finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 71.1.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 X))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 X)))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1)) (CAR X)), which finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 71.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS V W1)))) (CONS F (CONS (CAR D) (CONS (CAR Z) (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS V W1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1)) V), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CAR, EQUAL, and MEMBER, to: T. Case 70.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1)))) (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2)))))), which again simplifies, unfolding EQUAL, PLUS, and IFF, to: T. Case 69.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1)))) (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2)))))), which again simplifies, expanding EQUAL and IFF, to: T. Case 68.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1)))) F)), which again simplifies, rewriting with CAR-NLISTP, and opening up the functions EQUAL and MEMBER, to: T. Case 67.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1)))) F)). However this again simplifies, unfolding the functions EQUAL, PLUS, and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). The result is five new conjectures: Case 67.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and CAR, to: T. Case 67.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (CAR X) (NOT (CAR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). This further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, and MEMBER, to: T. Case 67.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X) (NOT V) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 67.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (NOT (CAR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). But this further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 67.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V (NOT X) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, applying CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 66.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1)))) F)). However this again simplifies, unfolding EQUAL, PLUS, and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We must thus prove seven new goals: Case 66.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, rewriting with the lemma CAR-NLISTP, and expanding STRIP-CARS, MEMBER, and CAR, to: T. Case 66.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X)) (NOT (CADR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 66.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X)) (NOT (CAR V)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 66.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X)) (NOT Z) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 66.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (NOT (CADR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). However this further simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, MEMBER, and CAR, to: T. Case 66.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (NOT (CAR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). This further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 66.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) (NOT Z) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NUMBERP (CDAR ARGS1))). However this further simplifies, applying CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 65.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1)))) F)). However this again simplifies, applying CAR-NLISTP, and expanding the definitions of EQUAL and MEMBER, to: T. Case 64.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1)))) F)). This again simplifies, unfolding EQUAL and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We must thus prove five new conjectures: Case 64.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))), which further simplifies, applying the lemmas CAR-NLISTP and CDR-NLISTP, and expanding STRIP-CARS, MEMBER, and CAR, to: T. Case 64.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (CAR X) (NOT (CAR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 64.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X) (NOT V) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 64.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (NOT (CAR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 64.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V (NOT X) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). This further simplifies, applying CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 63.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1)))) F)). But this again simplifies, expanding the functions EQUAL and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This generates seven new conjectures: Case 63.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))), which further simplifies, rewriting with CAR-NLISTP, and opening up STRIP-CARS, MEMBER, and CAR, to: T. Case 63.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X)) (NOT (CADR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, and MEMBER, to: T. Case 63.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X)) (NOT (CAR V)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). This further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: T. Case 63.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X)) (NOT Z) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 63.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (NOT (CADR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). This further simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, CONS, MEMBER, and CAR, to: T. Case 63.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (NOT (CAR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). However this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 63.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) (NOT Z) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). But this further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 62.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1)))) F)), which again simplifies, rewriting with CAR-NLISTP, and expanding EQUAL and MEMBER, to: T. Case 61.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1)))) F)). However this again simplifies, expanding the functions EQUAL, PLUS, and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We must thus prove five new formulas: Case 61.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, applying CAR-NLISTP and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 61.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (CAR X) (NOT (CAR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 61.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X) (NOT V) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 61.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (NOT (CAR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). This further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 61.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V (NOT X) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). This further simplifies, applying CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 60.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1)))) F)). But this again simplifies, opening up EQUAL, PLUS, and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We must thus prove seven new goals: Case 60.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, rewriting with CAR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and CAR, to: T. Case 60.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X)) (NOT (CADR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). But this further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 60.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X)) (NOT (CAR V)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 60.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X)) (NOT Z) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 60.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (NOT (CADR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, MEMBER, and CAR, to: T. Case 60.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (NOT (CAR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 60.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) (NOT Z) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NUMBERP (CDAR ARGS1))). However this further simplifies, applying CDR-CONS and CAR-CONS, and expanding STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 59.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1)))) F)). This again simplifies, applying CAR-NLISTP, and unfolding EQUAL and MEMBER, to: T. Case 58.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1)))) F)). This again simplifies, opening up EQUAL and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This generates five new conjectures: Case 58.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))), which further simplifies, applying CAR-NLISTP and CDR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and CAR, to: T. Case 58.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (CAR X) (NOT (CAR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 58.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X) (NOT V) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 58.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (NOT (CAR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). This further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 58.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V (NOT X) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 57.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1)))) F)). However this again simplifies, unfolding the definitions of EQUAL and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This generates seven new conjectures: Case 57.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))), which further simplifies, rewriting with the lemma CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 57.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X)) (NOT (CADR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 57.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X)) (NOT (CAR V)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and MEMBER, to: T. Case 57.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X)) (NOT Z) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 57.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (NOT (CADR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). However this further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding the functions STRIP-CARS, CONS, MEMBER, and CAR, to: T. Case 57.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (NOT (CAR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS and MEMBER, to: T. Case 57.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) (NOT Z) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1)) (NOT (NUMBERP (CDAR ARGS1)))). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 56.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1)))) (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2)))))). But this again simplifies, unfolding the definitions of EQUAL, PLUS, and IFF, to: T. Case 55.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1)))) (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2)))))), which again simplifies, expanding the functions EQUAL, PLUS, and IFF, to: T. Case 54.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAR ARGS1))) (IFF F (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2)))))), which again simplifies, rewriting with CAR-NLISTP, and opening up the functions EQUAL and MEMBER, to: T. Case 53.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1) (NOT (CADR ARGS1))) (IFF F (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2)))))). However this again simplifies, expanding EQUAL, PLUS, and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1)) (CADR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). The result is five new goals: Case 53.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1)) (CADR ARGS1)), which further simplifies, applying the lemmas CAR-NLISTP and CDR-NLISTP, and unfolding STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 53.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X))) (CAAR ARGS1)) (CADR ARGS1)), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding the definitions of STRIP-CARS, CONS, MEMBER, and NUMBERP, to: T. Case 53.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X))) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 53.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)), which further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS X Z1) to eliminate (CAR D) and (CDR D), and D by (CONS C X1) to eliminate (CAR D) and (CDR D). We thus obtain the following five new goals: Case 53.2.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). This finally simplifies, unfolding STRIP-CARS, to: T. Case 53.2.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NOT (NUMBERP W)) (CAR X)) (CAR D)), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS, CONS, CAR, EQUAL, and MEMBER, to: (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL 0 V) X (NOT (NUMBERP W))) 0), which finally simplifies, trivially, to: T. Case 53.2.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS C X1)))) V (NOT (NUMBERP W)) (CAR X)) C). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and CAR, to: T. Case 53.2.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NOT (NUMBERP W)) C) (CAR D)), which further simplifies, rewriting with CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and expanding STRIP-CARS, EQUAL, and MEMBER, to: (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL C V) (NOT (NUMBERP W)) C) 0), which finally simplifies, trivially, to: T. Case 53.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS X Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS X Z1)))) V (NOT (NUMBERP W)) C) X). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and CAR, to: T. Case 53.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). However this further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS C Z) to eliminate (CDR ARGS1) and (CAR ARGS1), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), Z by (CONS C V1) to eliminate (CAR Z) and (CDR Z), and Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z). We must thus prove five new goals: Case 53.1.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAAR ARGS1)) (CADR ARGS1)), which finally simplifies, opening up STRIP-CARS, to: T. Case 53.1.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C Z)) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C Z))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR C)) (CAR Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, CONS, and CAR, to: T. Case 53.1.3. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C (CONS X1 Z1))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C (CONS X1 Z1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR C)) X1). This finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 53.1.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) X1) (CAR Z)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definition of STRIP-CARS, to: T. Case 53.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS C V1))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS C V1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) X1) C), which finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 52.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (IFF F (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2)))))). This again simplifies, opening up EQUAL, PLUS, and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We must thus prove five new conjectures: Case 52.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, rewriting with the lemmas CAR-NLISTP and CDR-NLISTP, and unfolding STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 52.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, CONS, MEMBER, and NUMBERP, to: T. Case 52.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 52.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS Z1 X) to eliminate (CDR D) and (CAR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following seven new conjectures: Case 52.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). This finally simplifies, expanding STRIP-CARS, to: T. Case 52.2.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NOT (NUMBERP W)) (NOT (CAR X))) (CADR D)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS, CONS, CAR, EQUAL, and MEMBER, to: T. Case 52.2.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 C))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 C)))) V (NOT (NUMBERP W)) (NOT (CAR X))) (CAR C)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definition of STRIP-CARS, to: T. Case 52.2.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS D Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS D Z1))))) V (NOT (NUMBERP W)) (NOT (CAR X))) D). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and CAR, to: T. Case 52.2.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NOT (NUMBERP W)) (NOT C)) (CADR D)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and CAR, to: T. Case 52.2.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 X))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 X)))) V (NOT (NUMBERP W)) (NOT C)) (CAR X)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 52.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) V (NOT (NUMBERP W)) (NOT C)) D). This finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS and CAR, to: T. Case 52.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS C Z) to eliminate (CDR ARGS1) and (CAR ARGS1), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), Z by (CONS V1 C) to eliminate (CDR Z) and (CAR Z), C by (CONS Z W1) to eliminate (CAR C) and (CDR C), Z by (CONS Z1 X1) to eliminate (CDR Z) and (CAR Z), and X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1). We would thus like to prove the following seven new goals: Case 52.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this finally simplifies, opening up the function STRIP-CARS, to: T. Case 52.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C Z)) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C Z))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT (CAR C))) (CADR Z)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CONS, and CAR, to: T. Case 52.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C (CONS Z1 X1))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C (CONS Z1 X1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT (CAR C))) (CAR X1)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 52.1.4. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C (CONS Z1 (CONS Z V1)))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C (CONS Z1 (CONS Z V1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT (CAR C))) Z), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 52.1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1)) (CADR Z)). But this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding STRIP-CARS, CONS, and CAR, to: T. Case 52.1.2. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 C))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 C)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1)) (CAR C)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 52.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1)))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1)) Z). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and CAR, to: T. Case 51.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1)))) (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2)))))), which again simplifies, unfolding EQUAL, PLUS, and IFF, to: T. Case 50.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1)))) (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2)))))), which again simplifies, opening up the definitions of EQUAL, PLUS, and IFF, to: T. Case 49.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1)))) (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2)))))), which again simplifies, opening up the definitions of EQUAL, PLUS, and IFF, to: T. Case 48.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1)))) (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2)))))), which again simplifies, expanding the functions EQUAL and IFF, to: T. Case 47.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAR ARGS1))) (IFF F (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2)))))), which again simplifies, rewriting with the lemma CAR-NLISTP, and expanding the definitions of EQUAL and MEMBER, to: T. Case 46.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (IFF F (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2)))))), which again simplifies, opening up EQUAL and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1)) (CADR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We must thus prove five new conjectures: Case 46.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1)) (CADR ARGS1)), which further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to the new goal: (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X Z) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following five new conjectures: Case 46.5.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAAR ARGS1)) (CADR ARGS1)). However this further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and unfolding STRIP-CARS, EQUAL, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP ARGS2))) 0), which finally simplifies, trivially, to: T. Case 46.5.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X Z)) NIL) (NOT (MEMBER F (CONS X Z))) (CAR X)) (CAR Z)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 46.5.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS V W))) NIL) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X)) V). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and CAR, to: T. Case 46.5.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) Z)) NIL) (NOT (MEMBER F (CONS (CONS V W) Z))) V) (CAR Z)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 46.5.1. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS X D))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V) X). However this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and CAR, to: T. Case 46.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X)) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the functions STRIP-CARS, CONS, MEMBER, and NUMBERP, to the new formula: (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V W) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS D C) to eliminate (CAR V) and (CDR V), W by (CONS V X1) to eliminate (CAR W) and (CDR W), and W by (CONS D C) to eliminate (CAR W) and (CDR W). We would thus like to prove the following five new goals: Case 46.4.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAAR ARGS1)) (CADR ARGS1)). But this finally simplifies, opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 46.4.4. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V W)) '(0)) (NOT (MEMBER F (CONS V W))) X (CAR V)) (CAR W)), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, EQUAL, and MEMBER, to: (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) V X) 0), which finally simplifies, trivially, to: T. Case 46.4.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS D C))) '(0)) (NOT (MEMBER F (CONS V (CONS D C)))) X (CAR V)) D). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 46.4.2. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) W)) '(0)) (NOT (MEMBER F (CONS (CONS D C) W))) X D) (CAR W)). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and expanding STRIP-CARS, CAR, EQUAL, and MEMBER, to the new conjecture: (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL D 0) X) 0), which finally simplifies, obviously, to: T. Case 46.4.1. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS V X1))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS V X1)))) X D) V). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and CAR, to: T. Case 46.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X)) (CAAR ARGS1)) (CADR ARGS1)). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS D Z) to eliminate (CDR ARGS1) and (CAR ARGS1), D by (CONS C X1) to eliminate (CAR D) and (CDR D), Z by (CONS D Z1) to eliminate (CAR Z) and (CDR Z), and Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z). This produces the following five new conjectures: Case 46.3.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAAR ARGS1)) (CADR ARGS1)). However this finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 46.3.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS D Z)) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS D Z))) (NOT (MEMBER F W)) X V (CAR D)) (CAR Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 46.3.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS D (CONS C X1))) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS D (CONS C X1)))) (NOT (MEMBER F W)) X V (CAR D)) C). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CAR, EQUAL, and MEMBER, to: T. Case 46.3.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) Z))) (NOT (MEMBER F W)) X V C) (CAR Z)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 46.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS D Z1))) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS D Z1)))) (NOT (MEMBER F W)) X V C) D), which finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 46.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to the new goal: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS X Z1) to eliminate (CAR D) and (CDR D), and D by (CONS C X1) to eliminate (CAR D) and (CDR D). We thus obtain the following five new formulas: Case 46.2.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). But this finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 46.2.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NUMBERP W) (CAR X)) (CAR D)), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, CONS, CAR, EQUAL, and MEMBER, to: (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL 0 V) X (NUMBERP W)) 0), which finally simplifies, clearly, to: T. Case 46.2.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS C X1)))) V (NUMBERP W) (CAR X)) C). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS and CAR, to: T. Case 46.2.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NUMBERP W) C) (CAR D)), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and unfolding STRIP-CARS, EQUAL, and MEMBER, to: (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL C V) (NUMBERP W) C) 0). This finally simplifies, trivially, to: T. Case 46.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS X Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS X Z1)))) V (NUMBERP W) C) X). However this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS and CAR, to: T. Case 46.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). But this further simplifies, applying CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS C Z) to eliminate (CDR ARGS1) and (CAR ARGS1), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), Z by (CONS C V1) to eliminate (CAR Z) and (CDR Z), and Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following five new formulas: Case 46.1.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAAR ARGS1)) (CADR ARGS1)). This finally simplifies, unfolding STRIP-CARS, to: T. Case 46.1.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C Z)) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C Z))) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR C)) (CAR Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CONS, and CAR, to: T. Case 46.1.3. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C (CONS X1 Z1))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C (CONS X1 Z1)))) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR C)) X1). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 46.1.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) (NOT (MEMBER F D)) V X (NUMBERP W) X1) (CAR Z)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the function STRIP-CARS, to: T. Case 46.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS C V1))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS C V1)))) (NOT (MEMBER F D)) V X (NUMBERP W) X1) C). This finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 45.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (IFF F (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2)))))). This again simplifies, opening up the functions EQUAL and IFF, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We must thus prove five new goals: Case 45.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X Z) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We would thus like to prove the following seven new goals: Case 45.5.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this finally simplifies, rewriting with CAR-NLISTP, and expanding STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 45.5.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X Z)) NIL) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X))) (CADR Z)). But this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, and EQUAL, to: T. Case 45.5.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS W V))) NIL) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X))) (CAR V)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, to: T. Case 45.5.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS W (CONS Z D)))) NIL) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X))) Z). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and CAR, to: T. Case 45.5.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) Z)) NIL) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V)) (CADR Z)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 45.5.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS D X))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V)) (CAR X)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 45.5.1. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C)))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V)) Z). However this finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and CAR, to: T. Case 45.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X)) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, MEMBER, and NUMBERP, to the new formula: (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V W) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS D C) to eliminate (CAR V) and (CDR V), W by (CONS X1 V) to eliminate (CDR W) and (CAR W), V by (CONS W Z1) to eliminate (CAR V) and (CDR V), W by (CONS C D) to eliminate (CDR W) and (CAR W), and D by (CONS W X1) to eliminate (CAR D) and (CDR D). We thus obtain the following seven new conjectures: Case 45.4.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (NOT (CAAR ARGS1))) (CADDR ARGS1)). However this finally simplifies, expanding STRIP-CARS and EQUAL, to: T. Case 45.4.6. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V W)) '(0)) (NOT (MEMBER F (CONS V W))) X (NOT (CAR V))) (CADR W)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, EQUAL, and MEMBER, to: T. Case 45.4.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C D))) '(0)) (NOT (MEMBER F (CONS V (CONS C D)))) X (NOT (CAR V))) (CAR D)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 45.4.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C (CONS W X1)))) '(0)) (NOT (MEMBER F (CONS V (CONS C (CONS W X1))))) X (NOT (CAR V))) W). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 45.4.3. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) W)) '(0)) (NOT (MEMBER F (CONS (CONS D C) W))) X (NOT D)) (CADR W)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CONS, and EQUAL, to: T. Case 45.4.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 V))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 V)))) X (NOT D)) (CAR V)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 45.4.1. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS W Z1)))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS W Z1))))) X (NOT D)) W). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 45.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X)) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and NUMBERP, to the formula: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (NOT (CAAR ARGS1))) (CADDR ARGS1)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS D Z) to eliminate (CDR ARGS1) and (CAR ARGS1), D by (CONS C X1) to eliminate (CAR D) and (CDR D), Z by (CONS Z1 D) to eliminate (CDR Z) and (CAR Z), D by (CONS Z V1) to eliminate (CAR D) and (CDR D), Z by (CONS X1 C) to eliminate (CDR Z) and (CAR Z), and C by (CONS Z Z1) to eliminate (CAR C) and (CDR C). We must thus prove seven new formulas: Case 45.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (NOT (CAAR ARGS1))) (CADDR ARGS1)), which finally simplifies, opening up the function STRIP-CARS, to: T. Case 45.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS D Z)) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS D Z))) (NOT (MEMBER F W)) X V (NOT (CAR D))) (CADR Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 45.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS D (CONS X1 C))) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS D (CONS X1 C)))) (NOT (MEMBER F W)) X V (NOT (CAR D))) (CAR C)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 45.3.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS D (CONS X1 (CONS Z Z1)))) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS D (CONS X1 (CONS Z Z1))))) (NOT (MEMBER F W)) X V (NOT (CAR D))) Z). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CAR, EQUAL, and MEMBER, to: T. Case 45.3.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) Z))) (NOT (MEMBER F W)) X V (NOT C)) (CADR Z)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 45.3.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 D))) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 D)))) (NOT (MEMBER F W)) X V (NOT C)) (CAR D)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 45.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS Z V1)))) (CONS 0 (CONS (CAR V) (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS Z V1))))) (NOT (MEMBER F W)) X V (NOT C)) Z). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CAR, and EQUAL, to: T. Case 45.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS Z1 X) to eliminate (CDR D) and (CAR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). This produces the following seven new formulas: Case 45.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). But this finally simplifies, unfolding the function STRIP-CARS, to: T. Case 45.2.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NUMBERP W) (NOT (CAR X))) (CADR D)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CONS, CAR, EQUAL, and MEMBER, to: T. Case 45.2.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 C))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 C)))) V (NUMBERP W) (NOT (CAR X))) (CAR C)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the function STRIP-CARS, to: T. Case 45.2.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS D Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS D Z1))))) V (NUMBERP W) (NOT (CAR X))) D), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS and CAR, to: T. Case 45.2.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NUMBERP W) (NOT C)) (CADR D)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding STRIP-CARS, CONS, and CAR, to: T. Case 45.2.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 X))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 X)))) V (NUMBERP W) (NOT C)) (CAR X)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 45.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) V (NUMBERP W) (NOT C)) D). However this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding STRIP-CARS and CAR, to: T. Case 45.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). This further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS C Z) to eliminate (CDR ARGS1) and (CAR ARGS1), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), Z by (CONS V1 C) to eliminate (CDR Z) and (CAR Z), C by (CONS Z W1) to eliminate (CAR C) and (CDR C), Z by (CONS Z1 X1) to eliminate (CDR Z) and (CAR Z), and X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1). We would thus like to prove the following seven new conjectures: Case 45.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT (CAAR ARGS1))) (CADDR ARGS1)). But this finally simplifies, expanding the function STRIP-CARS, to: T. Case 45.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C Z)) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C Z))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT (CAR C))) (CADR Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 45.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C (CONS Z1 X1))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C (CONS Z1 X1)))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT (CAR C))) (CAR X1)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS and MEMBER, to: T. Case 45.1.4. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS C (CONS Z1 (CONS Z V1)))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS C (CONS Z1 (CONS Z V1))))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT (CAR C))) Z). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 45.1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1)) (CADR Z)). But this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CONS, and CAR, to: T. Case 45.1.2. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 C))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 C)))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1)) (CAR C)), which finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 45.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1)))) (CONS V (CONS (CAR X) (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1))))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1)) Z), which finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS and CAR, to: T. Case 44.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1)))) (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2)))))), which again simplifies, expanding the functions EQUAL, PLUS, and IFF, to: T. Case 43.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (IFF (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1)))) (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2)))))), which again simplifies, unfolding the definitions of EQUAL and IFF, to: T. Case 42.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (EQUAL FN 'IF)) (SUBRP FN)) (EQUAL (CAR (CONS (APPLY-SUBR FN (STRIP-CARS ARGS1)) (ADD1 (SUM-CDRS ARGS1)))) (CAR (CONS (APPLY-SUBR FN (STRIP-CARS ARGS1)) (ADD1 (SUM-CDRS ARGS2)))))), which again simplifies, appealing to the lemma CAR-CONS, to: T. Case 41.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (EQUAL FN 'IF)) (NOT (SUBRP FN)) (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1)))) (EQUAL (CAR (CONS (CAR (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1)))) (ADD1 (PLUS (SUM-CDRS ARGS1) (CDR (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1)))))))) (CAR (CONS (CAR (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1)))) (ADD1 (PLUS (SUM-CDRS ARGS2) (CDR (V&C\$ T (BODY FN) (PAIRLIST (FORMALS FN) (STRIP-CARS ARGS1)))))))))), which again simplifies, applying CAR-CONS, to: T. Case 40.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1))))) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2))))))). This again simplifies, applying CAR-CONS, and expanding the definitions of EQUAL and PLUS, to the new formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We thus obtain the following nine new formulas: Case 40.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADDR ARGS2))). However this further simplifies, applying the lemma CAR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 40.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADR Z))), which further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 40.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAAR V))), which further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 40.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAR Z))), which further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 40.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) V)), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 40.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADR Z))). This further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X1 X) to eliminate (CDR V) and (CAR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS X V1) to eliminate (CAR V) and (CDR V), V by (CONS C D) to eliminate (CDR V) and (CAR V), D by (CONS V X1) to eliminate (CAR D) and (CDR D), and V by (CONS D Z1) to eliminate (CAR V) and (CDR V). We would thus like to prove the following nine new conjectures: Case 40.4.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). However this finally simplifies, unfolding STRIP-CARS and EQUAL, to: T. Case 40.4.8. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NOT (NUMBERP W)) X (NOT (CAR X)) (CADR V) (NOT (NUMBERP (CDR X)))) (EQUAL (CAADR V) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and EQUAL, to: T. Case 40.4.7. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C D))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C D)))) (NOT (NUMBERP W)) X (NOT (CAR X)) (CAR D) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR D) 0)). This finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 40.4.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS V X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS V X1))))) (NOT (NUMBERP W)) X (NOT (CAR X)) V (NOT (NUMBERP (CDR X)))) (EQUAL (CAR V) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 40.4.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS (CONS D Z1) X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS (CONS D Z1) X1))))) (NOT (NUMBERP W)) X (NOT (CAR X)) (NOT (NUMBERP (CDR X)))) (EQUAL D 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CAR, and EQUAL, to: T. Case 40.4.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NOT (NUMBERP W)) (NOT D) (CADR V) (NOT (NUMBERP C))) (EQUAL (CAADR V) 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, EQUAL, MEMBER, and CAR, to: T. Case 40.4.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 X))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 X)))) (NOT (NUMBERP W)) (NOT D) (CAR X) (NOT (NUMBERP C))) (EQUAL (CAAR X) 0)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 40.4.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS V Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS V Z1))))) (NOT (NUMBERP W)) (NOT D) V (NOT (NUMBERP C))) (EQUAL (CAR V) 0)). This finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 40.4.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS (CONS X V1) Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS (CONS X V1) Z1))))) (NOT (NUMBERP W)) (NOT D) (NOT (NUMBERP C))) (EQUAL X 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 40.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAAR X))). However this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and CAR, to the new formula: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS Z1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS V W1) to eliminate (CAR Z) and (CDR Z), Z by (CONS X1 C) to eliminate (CDR Z) and (CAR Z), C by (CONS Z Z1) to eliminate (CAR C) and (CDR C), and Z by (CONS C V1) to eliminate (CAR Z) and (CDR Z). This produces the following nine new formulas: Case 40.3.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). This finally simplifies, unfolding the function STRIP-CARS, to: T. Case 40.3.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NOT (NUMBERP W)) V (NOT (CAR V)) (CADR Z) (NOT (NUMBERP (CDR V)))) (EQUAL (CAADR Z) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 40.3.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 C)))) D (NOT (NUMBERP W)) V (NOT (CAR V)) (CAR C) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR C) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 40.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS Z Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS Z Z1))))) D (NOT (NUMBERP W)) V (NOT (CAR V)) Z (NOT (NUMBERP (CDR V)))) (EQUAL (CAR Z) 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 40.3.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS (CONS C V1) Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS (CONS C V1) Z1))))) D (NOT (NUMBERP W)) V (NOT (CAR V)) (NOT (NUMBERP (CDR V)))) (EQUAL C 0)). But this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 40.3.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NOT (NUMBERP W)) (NOT C) (CADR Z) (NOT (NUMBERP X1))) (EQUAL (CAADR Z) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 40.3.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) D (NOT (NUMBERP W)) (NOT C) (CAR V) (NOT (NUMBERP X1))) (EQUAL (CAAR V) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and unfolding STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 40.3.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS Z V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS Z V1))))) D (NOT (NUMBERP W)) (NOT C) Z (NOT (NUMBERP X1))) (EQUAL (CAR Z) 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 40.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1))))) D (NOT (NUMBERP W)) (NOT C) (NOT (NUMBERP X1))) (EQUAL V 0)). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 40.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAR Z))). This further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS and MEMBER, to the new formula: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V X) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), X by (CONS V1 V) to eliminate (CDR X) and (CAR X), V by (CONS X W1) to eliminate (CAR V) and (CDR V), X by (CONS V D1) to eliminate (CAR X) and (CDR X), X by (CONS Z1 X1) to eliminate (CDR X) and (CAR X), X1 by (CONS X V1) to eliminate (CAR X1) and (CDR X1), and X by (CONS X1 W1) to eliminate (CAR X) and (CDR X). We thus obtain the following nine new goals: Case 40.2.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). But this finally simplifies, expanding STRIP-CARS, to: T. Case 40.2.8. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V X))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (NOT (CAR V)) (CADR X) (NOT (NUMBERP (CDR V)))) (EQUAL (CAADR X) 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 40.2.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (NOT (CAR V)) (CAR X1) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR X1) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 40.2.6. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS X V1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS X V1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (NOT (CAR V)) X (NOT (NUMBERP (CDR V)))) (EQUAL (CAR X) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 40.2.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS (CONS X1 W1) V1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (NOT (CAR V)) (NOT (NUMBERP (CDR V)))) (EQUAL X1 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 40.2.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) X))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1) (CADR X) (NOT (NUMBERP Z1))) (EQUAL (CAADR X) 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 40.2.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1) (CAR V) (NOT (NUMBERP Z1))) (EQUAL (CAAR V) 0)). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 40.2.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS X W1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS X W1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1) X (NOT (NUMBERP Z1))) (EQUAL (CAR X) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 40.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1) (NOT (NUMBERP Z1))) (EQUAL V 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 40.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) X)). This further simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) X)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS V D1) to eliminate (CAR Z) and (CDR Z), Z by (CONS Z1 X1) to eliminate (CDR Z) and (CAR Z), X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1), and Z by (CONS X1 W1) to eliminate (CAR Z) and (CDR Z). We must thus prove nine new conjectures: Case 40.1.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) X)), which finally simplifies, expanding STRIP-CARS, to: T. Case 40.1.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (NOT (CAR V)) (CADR Z) (NOT (NUMBERP (CDR V)))) (EQUAL (CAADR Z) X)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 40.1.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (NOT (CAR V)) (CAR X1) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR X1) X)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 40.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS Z V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS Z V1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (NOT (CAR V)) Z (NOT (NUMBERP (CDR V)))) (EQUAL (CAR Z) X)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 40.1.5. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS (CONS X1 W1) V1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (NOT (CAR V)) (NOT (NUMBERP (CDR V)))) (EQUAL X1 X)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and EQUAL, to: T. Case 40.1.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (CADR Z) (NOT (NUMBERP Z1))) (EQUAL (CAADR Z) X)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 40.1.3. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (CAR V) (NOT (NUMBERP Z1))) (EQUAL (CAAR V) X)). However this finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS and EQUAL, to: T. Case 40.1.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) Z (NOT (NUMBERP Z1))) (EQUAL (CAR Z) X)), which finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 40.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (NOT (NUMBERP Z1))) (EQUAL V X)). However this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 39.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1))))) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2))))))). But this again simplifies, applying CAR-CONS, and unfolding EQUAL and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We thus obtain the following nine new formulas: Case 39.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADDR ARGS2))). This further simplifies, applying CAR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and CAR, to: T. Case 39.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADR Z))). This further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 39.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAAR V))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 39.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAR Z))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 39.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) V)). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 39.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADR Z))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, CONS, MEMBER, and CAR, to the new formula: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X1 X) to eliminate (CDR V) and (CAR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS X V1) to eliminate (CAR V) and (CDR V), V by (CONS C D) to eliminate (CDR V) and (CAR V), D by (CONS V X1) to eliminate (CAR D) and (CDR D), and V by (CONS D Z1) to eliminate (CAR V) and (CDR V). We thus obtain the following nine new formulas: Case 39.4.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). But this finally simplifies, expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 39.4.8. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NOT (NUMBERP W)) X (NOT (CAR X)) (CADR V) (NUMBERP (CDR X))) (EQUAL (CAADR V) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, and EQUAL, to: T. Case 39.4.7. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C D))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C D)))) (NOT (NUMBERP W)) X (NOT (CAR X)) (CAR D) (NUMBERP (CDR X))) (EQUAL (CAAR D) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 39.4.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS V X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS V X1))))) (NOT (NUMBERP W)) X (NOT (CAR X)) V (NUMBERP (CDR X))) (EQUAL (CAR V) 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CAR, and EQUAL, to: T. Case 39.4.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS (CONS D Z1) X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS (CONS D Z1) X1))))) (NOT (NUMBERP W)) X (NOT (CAR X)) (NUMBERP (CDR X))) (EQUAL D 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 39.4.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NOT (NUMBERP W)) (NOT D) (CADR V) (NUMBERP C)) (EQUAL (CAADR V) 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, CONS, EQUAL, MEMBER, and CAR, to: T. Case 39.4.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 X))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 X)))) (NOT (NUMBERP W)) (NOT D) (CAR X) (NUMBERP C)) (EQUAL (CAAR X) 0)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 39.4.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS V Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS V Z1))))) (NOT (NUMBERP W)) (NOT D) V (NUMBERP C)) (EQUAL (CAR V) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 39.4.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS (CONS X V1) Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS (CONS X V1) Z1))))) (NOT (NUMBERP W)) (NOT D) (NUMBERP C)) (EQUAL X 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 39.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAAR X))), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and CAR, to the new conjecture: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS Z1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS V W1) to eliminate (CAR Z) and (CDR Z), Z by (CONS X1 C) to eliminate (CDR Z) and (CAR Z), C by (CONS Z Z1) to eliminate (CAR C) and (CDR C), and Z by (CONS C V1) to eliminate (CAR Z) and (CDR Z). We thus obtain the following nine new goals: Case 39.3.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). But this finally simplifies, expanding the function STRIP-CARS, to: T. Case 39.3.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NOT (NUMBERP W)) V (NOT (CAR V)) (CADR Z) (NUMBERP (CDR V))) (EQUAL (CAADR Z) 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 39.3.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 C)))) D (NOT (NUMBERP W)) V (NOT (CAR V)) (CAR C) (NUMBERP (CDR V))) (EQUAL (CAAR C) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 39.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS Z Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS Z Z1))))) D (NOT (NUMBERP W)) V (NOT (CAR V)) Z (NUMBERP (CDR V))) (EQUAL (CAR Z) 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 39.3.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS (CONS C V1) Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS (CONS C V1) Z1))))) D (NOT (NUMBERP W)) V (NOT (CAR V)) (NUMBERP (CDR V))) (EQUAL C 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 39.3.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NOT (NUMBERP W)) (NOT C) (CADR Z) (NUMBERP X1)) (EQUAL (CAADR Z) 0)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 39.3.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) D (NOT (NUMBERP W)) (NOT C) (CAR V) (NUMBERP X1)) (EQUAL (CAAR V) 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and opening up the functions STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 39.3.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS Z V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS Z V1))))) D (NOT (NUMBERP W)) (NOT C) Z (NUMBERP X1)) (EQUAL (CAR Z) 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 39.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1))))) D (NOT (NUMBERP W)) (NOT C) (NUMBERP X1)) (EQUAL V 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 39.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAR Z))). However this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V X) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), X by (CONS V1 V) to eliminate (CDR X) and (CAR X), V by (CONS X W1) to eliminate (CAR V) and (CDR V), X by (CONS V D1) to eliminate (CAR X) and (CDR X), X by (CONS Z1 X1) to eliminate (CDR X) and (CAR X), X1 by (CONS X V1) to eliminate (CAR X1) and (CDR X1), and X by (CONS X1 W1) to eliminate (CAR X) and (CDR X). This produces the following nine new goals: Case 39.2.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). But this finally simplifies, unfolding the function STRIP-CARS, to: T. Case 39.2.8. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V X))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (NOT (CAR V)) (CADR X) (NUMBERP (CDR V))) (EQUAL (CAADR X) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 39.2.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (NOT (CAR V)) (CAR X1) (NUMBERP (CDR V))) (EQUAL (CAAR X1) 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 39.2.6. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS X V1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS X V1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (NOT (CAR V)) X (NUMBERP (CDR V))) (EQUAL (CAR X) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 39.2.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS (CONS X1 W1) V1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (NOT (CAR V)) (NUMBERP (CDR V))) (EQUAL X1 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 39.2.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) X))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1) (CADR X) (NUMBERP Z1)) (EQUAL (CAADR X) 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 39.2.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1) (CAR V) (NUMBERP Z1)) (EQUAL (CAAR V) 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 39.2.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS X W1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS X W1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1) X (NUMBERP Z1)) (EQUAL (CAR X) 0)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS, EQUAL, and MEMBER, to: T. Case 39.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1))))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (NOT X1) (NUMBERP Z1)) (EQUAL V 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 39.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) X)), which further simplifies, applying CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) X)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS V D1) to eliminate (CAR Z) and (CDR Z), Z by (CONS Z1 X1) to eliminate (CDR Z) and (CAR Z), X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1), and Z by (CONS X1 W1) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following nine new formulas: Case 39.1.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) X)). But this finally simplifies, opening up STRIP-CARS, to: T. Case 39.1.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (NOT (CAR V)) (CADR Z) (NUMBERP (CDR V))) (EQUAL (CAADR Z) X)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 39.1.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (NOT (CAR V)) (CAR X1) (NUMBERP (CDR V))) (EQUAL (CAAR X1) X)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 39.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS Z V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS Z V1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (NOT (CAR V)) Z (NUMBERP (CDR V))) (EQUAL (CAR Z) X)). However this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 39.1.5. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS (CONS X1 W1) V1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (NOT (CAR V)) (NUMBERP (CDR V))) (EQUAL X1 X)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 39.1.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (CADR Z) (NUMBERP Z1)) (EQUAL (CAADR Z) X)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 39.1.3. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (CAR V) (NUMBERP Z1)) (EQUAL (CAAR V) X)). But this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS and EQUAL, to: T. Case 39.1.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) Z (NUMBERP Z1)) (EQUAL (CAR Z) X)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 39.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (NUMBERP Z1)) (EQUAL V X)), which finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS and EQUAL, to: T. Case 38.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2))))))), which again simplifies, rewriting with CAR-NLISTP, and unfolding the definitions of EQUAL and MEMBER, to: T. Case 37.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2))))))). However this again simplifies, applying CAR-CONS, and expanding the definitions of EQUAL, CAR, and PLUS, to the new goal: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). This produces the following nine new conjectures: Case 37.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADDR ARGS2))). However this further simplifies, applying CAR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and CAR, to: T. Case 37.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADR Z))). But this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CONS, and MEMBER, to: T. Case 37.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAAR V))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and MEMBER, to: T. Case 37.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAR Z))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 37.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NOT (NUMBERP (CDR X))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 V)). But this further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 37.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADR Z))), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, CONS, MEMBER, CAR, and EQUAL, to: T. Case 37.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAAR X))). This further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, CAR, and EQUAL, to: T. Case 37.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAR Z))). This further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and EQUAL, to: T. Case 37.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 X)). But this further simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS and MEMBER, to the formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 X)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V V1) to eliminate (CAR Z) and (CDR Z), and Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z). This generates five new conjectures: Case 37.1.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 X)), which finally simplifies, unfolding STRIP-CARS, to: T. Case 37.1.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR V) (NOT (CAR Z))) (EQUAL 0 X)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 37.1.3. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR V) (NOT X1)) (EQUAL 0 X)). This finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CAR, and EQUAL, to: T. Case 37.1.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) X1 (NOT (CAR Z))) (EQUAL 0 X)), which finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 37.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) X1 (NOT V)) (EQUAL 0 X)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS and CAR, to: T. Case 36.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2))))))). But this again simplifies, applying CAR-CONS, and expanding EQUAL, CAR, and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We thus obtain the following nine new formulas: Case 36.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADDR ARGS2))). But this further simplifies, rewriting with CAR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and CAR, to: T. Case 36.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADR Z))). But this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, and MEMBER, to: T. Case 36.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAAR V))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: T. Case 36.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAR Z))). But this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 36.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 V)), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 36.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADR Z))). However this further simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, MEMBER, CAR, and EQUAL, to: T. Case 36.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAAR X))). But this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up STRIP-CARS, MEMBER, CAR, and EQUAL, to: T. Case 36.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAR Z))). This further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and EQUAL, to: T. Case 36.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 X)), which further simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 X)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS Z1 X1) to eliminate (CDR Z) and (CAR Z), and X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1). We thus obtain the following seven new conjectures: Case 36.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 X)). This finally simplifies, unfolding the definition of STRIP-CARS, to: T. Case 36.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT (CAR V)) (NOT (CADR Z))) (EQUAL 0 X)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 36.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT (CAR V)) (NOT (CAR X1))) (EQUAL 0 X)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 36.1.4. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS Z V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS Z V1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT (CAR V)) (NOT Z)) (EQUAL 0 X)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 36.1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (NOT (CADR Z))) (EQUAL 0 X)). This finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 36.1.2. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (NOT (CAR V))) (EQUAL 0 X)), which finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS and EQUAL, to: T. Case 36.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1))))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (NOT X1) (NOT Z)) (EQUAL 0 X)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 35.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1))))) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2))))))), which again simplifies, applying the lemma CAR-CONS, and opening up the definitions of EQUAL and PLUS, to the formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADDR ARGS2))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We must thus prove nine new formulas: Case 35.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADDR ARGS2))), which further simplifies, appealing to the lemma CAR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 35.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADR Z))), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 35.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAAR V))), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 35.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAR Z))). This further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 35.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) V)), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: T. Case 35.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADR Z))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding the functions STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X X1) to eliminate (CAR V) and (CDR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS D C) to eliminate (CAR V) and (CDR V), and D by (CONS V X1) to eliminate (CAR D) and (CDR D). We would thus like to prove the following seven new goals: Case 35.4.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). But this finally simplifies, unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 35.4.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NOT (NUMBERP W)) X (CAR X) (CAR V) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR V) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, and EQUAL, to: T. Case 35.4.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS D C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS D C)))) (NOT (NUMBERP W)) X (CAR X) D (NOT (NUMBERP (CDR X)))) (EQUAL (CAR D) 0)). This finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 35.4.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS V X1) C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS (CONS V X1) C)))) (NOT (NUMBERP W)) X (CAR X) (NOT (NUMBERP (CDR X)))) (EQUAL V 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CAR, and EQUAL, to: T. Case 35.4.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NOT (NUMBERP W)) D (CAR V) (NOT (NUMBERP C))) (EQUAL (CAAR V) 0)), which finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and CAR, to: T. Case 35.4.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X X1)))) (NOT (NUMBERP W)) D X (NOT (NUMBERP C))) (EQUAL (CAR X) 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and CAR, to: T. Case 35.4.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS (CONS V Z1) X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS (CONS V Z1) X1)))) (NOT (NUMBERP W)) D (NOT (NUMBERP C))) (EQUAL V 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS and CAR, to: T. Case 35.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAAR X))). This further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS V Z1) to eliminate (CAR Z) and (CDR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z), and C by (CONS Z Z1) to eliminate (CAR C) and (CDR C). This produces the following seven new formulas: Case 35.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). However this finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 35.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NOT (NUMBERP W)) V (CAR V) (CAR Z) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR Z) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 35.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS C X1)))) D (NOT (NUMBERP W)) V (CAR V) C (NOT (NUMBERP (CDR V)))) (EQUAL (CAR C) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and EQUAL, to: T. Case 35.3.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS Z Z1) X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS (CONS Z Z1) X1)))) D (NOT (NUMBERP W)) V (CAR V) (NOT (NUMBERP (CDR V)))) (EQUAL Z 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 35.3.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NOT (NUMBERP W)) C (CAR Z) (NOT (NUMBERP X1))) (EQUAL (CAAR Z) 0)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding STRIP-CARS, to: T. Case 35.3.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) D (NOT (NUMBERP W)) C V (NOT (NUMBERP X1))) (EQUAL (CAR V) 0)). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS, to: T. Case 35.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS Z V1) Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS Z V1) Z1)))) D (NOT (NUMBERP W)) C (NOT (NUMBERP X1))) (EQUAL Z 0)). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 35.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAR Z))), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V X) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), X by (CONS V V1) to eliminate (CAR X) and (CDR X), V by (CONS X W1) to eliminate (CAR V) and (CDR V), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), and X1 by (CONS X V1) to eliminate (CAR X1) and (CDR X1). We thus obtain the following seven new goals: Case 35.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 35.2.6. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V X))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (CAR V) (CAR X) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR X) 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 35.2.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (CAR V) X1 (NOT (NUMBERP (CDR V)))) (EQUAL (CAR X1) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 35.2.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS X V1) Z1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS (CONS X V1) Z1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (CAR V) (NOT (NUMBERP (CDR V)))) (EQUAL X 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 35.2.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) X))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) X1 (CAR X) (NOT (NUMBERP Z1))) (EQUAL (CAAR X) 0)). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 35.2.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) X1 V (NOT (NUMBERP Z1))) (EQUAL (CAR V) 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definition of STRIP-CARS, to: T. Case 35.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS X W1) V1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS X W1) V1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) X1 (NOT (NUMBERP Z1))) (EQUAL X 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 35.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) X)). But this further simplifies, applying CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) X)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V V1) to eliminate (CAR Z) and (CDR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), and X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1). This produces the following seven new goals: Case 35.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) X)). However this finally simplifies, unfolding the function STRIP-CARS, to: T. Case 35.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (CAR V) (CAR Z) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR Z) X)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 35.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (CAR V) X1 (NOT (NUMBERP (CDR V)))) (EQUAL (CAR X1) X)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 35.1.4. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS Z V1) Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS (CONS Z V1) Z1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (CAR V) (NOT (NUMBERP (CDR V)))) (EQUAL Z X)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 35.1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) X1 (CAR Z) (NOT (NUMBERP Z1))) (EQUAL (CAAR Z) X)), which finally simplifies, applying CDR-CONS and CAR-CONS, and expanding STRIP-CARS, to: T. Case 35.1.2. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) X1 V (NOT (NUMBERP Z1))) (EQUAL (CAR V) X)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definition of STRIP-CARS, to: T. Case 35.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS Z W1) V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS Z W1) V1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) X1 (NOT (NUMBERP Z1))) (EQUAL Z X)), which finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 34.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1))))) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS 0 (CDADDR ARGS2))))))). But this again simplifies, applying CAR-CONS, and expanding EQUAL and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). This produces the following nine new conjectures: Case 34.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADDR ARGS2))). This further simplifies, applying CAR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 34.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADR Z))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, and MEMBER, to: T. Case 34.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAAR V))). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and MEMBER, to: T. Case 34.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAR Z))). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and MEMBER, to: T. Case 34.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) V)). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 34.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADR Z))). However this further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, CONS, MEMBER, and CAR, to the goal: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X X1) to eliminate (CAR V) and (CDR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS D C) to eliminate (CAR V) and (CDR V), and D by (CONS V X1) to eliminate (CAR D) and (CDR D). This generates seven new goals: Case 34.4.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)), which finally simplifies, expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 34.4.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NOT (NUMBERP W)) X (CAR X) (CAR V) (NUMBERP (CDR X))) (EQUAL (CAAR V) 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, and EQUAL, to: T. Case 34.4.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS D C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS D C)))) (NOT (NUMBERP W)) X (CAR X) D (NUMBERP (CDR X))) (EQUAL (CAR D) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CAR, and EQUAL, to: T. Case 34.4.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS V X1) C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS (CONS V X1) C)))) (NOT (NUMBERP W)) X (CAR X) (NUMBERP (CDR X))) (EQUAL V 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 34.4.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NOT (NUMBERP W)) D (CAR V) (NUMBERP C)) (EQUAL (CAAR V) 0)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up STRIP-CARS and CAR, to: T. Case 34.4.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X X1)))) (NOT (NUMBERP W)) D X (NUMBERP C)) (EQUAL (CAR X) 0)). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and CAR, to: T. Case 34.4.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS (CONS V Z1) X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS (CONS V Z1) X1)))) (NOT (NUMBERP W)) D (NUMBERP C)) (EQUAL V 0)). This finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS and CAR, to: T. Case 34.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAAR X))), which further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up STRIP-CARS, MEMBER, and CAR, to the formula: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS V Z1) to eliminate (CAR Z) and (CDR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z), and C by (CONS Z Z1) to eliminate (CAR C) and (CDR C). The result is seven new conjectures: Case 34.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)), which finally simplifies, unfolding the function STRIP-CARS, to: T. Case 34.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NOT (NUMBERP W)) V (CAR V) (CAR Z) (NUMBERP (CDR V))) (EQUAL (CAAR Z) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 34.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS C X1)))) D (NOT (NUMBERP W)) V (CAR V) C (NUMBERP (CDR V))) (EQUAL (CAR C) 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 34.3.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS Z Z1) X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS (CONS Z Z1) X1)))) D (NOT (NUMBERP W)) V (CAR V) (NUMBERP (CDR V))) (EQUAL Z 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 34.3.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NOT (NUMBERP W)) C (CAR Z) (NUMBERP X1)) (EQUAL (CAAR Z) 0)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 34.3.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) D (NOT (NUMBERP W)) C V (NUMBERP X1)) (EQUAL (CAR V) 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS, to: T. Case 34.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS Z V1) Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS Z V1) Z1)))) D (NOT (NUMBERP W)) C (NUMBERP X1)) (EQUAL Z 0)). This finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 34.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAR Z))), which further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V X) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), X by (CONS V V1) to eliminate (CAR X) and (CDR X), V by (CONS X W1) to eliminate (CAR V) and (CDR V), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), and X1 by (CONS X V1) to eliminate (CAR X1) and (CDR X1). We would thus like to prove the following seven new formulas: Case 34.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). This finally simplifies, opening up the function STRIP-CARS, to: T. Case 34.2.6. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V X))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (CAR V) (CAR X) (NUMBERP (CDR V))) (EQUAL (CAAR X) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 34.2.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (CAR V) X1 (NUMBERP (CDR V))) (EQUAL (CAR X1) 0)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 34.2.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS X V1) Z1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS (CONS X V1) Z1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) V (CAR V) (NUMBERP (CDR V))) (EQUAL X 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 34.2.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) X))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) X1 (CAR X) (NUMBERP Z1)) (EQUAL (CAAR X) 0)). This finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 34.2.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) X1 V (NUMBERP Z1)) (EQUAL (CAR V) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definition of STRIP-CARS, to: T. Case 34.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS X W1) V1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS X W1) V1)))) D (NOT (MEMBER F C)) Z (NOT (NUMBERP W)) X1 (NUMBERP Z1)) (EQUAL X 0)), which finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 34.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) X)), which further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding STRIP-CARS and MEMBER, to the new goal: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) X)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V V1) to eliminate (CAR Z) and (CDR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), and X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1). This produces the following seven new conjectures: Case 34.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) X)). However this finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 34.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (CAR V) (CAR Z) (NUMBERP (CDR V))) (EQUAL (CAAR Z) X)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 34.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (CAR V) X1 (NUMBERP (CDR V))) (EQUAL (CAR X1) X)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 34.1.4. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS Z V1) Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS (CONS Z V1) Z1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) V (CAR V) (NUMBERP (CDR V))) (EQUAL Z X)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 34.1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) X1 (CAR Z) (NUMBERP Z1)) (EQUAL (CAAR Z) X)). But this finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and opening up the definition of STRIP-CARS, to: T. Case 34.1.2. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) X1 V (NUMBERP Z1)) (EQUAL (CAR V) X)), which finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up STRIP-CARS, to: T. Case 34.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS Z W1) V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS Z W1) V1)))) D (NOT (MEMBER F C)) (NOT (NUMBERP W)) X1 (NUMBERP Z1)) (EQUAL Z X)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 33.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1))))) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2))))))). However this again simplifies, rewriting with CAR-CONS, and expanding the definitions of EQUAL and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). This produces the following nine new formulas: Case 33.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADDR ARGS2))). This further simplifies, rewriting with CAR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 33.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADR Z))). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 33.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAAR V))). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 33.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAR Z))). This further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 33.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) V)), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 33.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADR Z))). But this further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X1 X) to eliminate (CDR V) and (CAR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS X V1) to eliminate (CAR V) and (CDR V), V by (CONS C D) to eliminate (CDR V) and (CAR V), D by (CONS V X1) to eliminate (CAR D) and (CDR D), and V by (CONS D Z1) to eliminate (CAR V) and (CDR V). The result is nine new formulas: Case 33.4.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 33.4.8. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NUMBERP W) X (NOT (CAR X)) (CADR V) (NOT (NUMBERP (CDR X)))) (EQUAL (CAADR V) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 33.4.7. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C D))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C D)))) (NUMBERP W) X (NOT (CAR X)) (CAR D) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR D) 0)). This finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 33.4.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS V X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS V X1))))) (NUMBERP W) X (NOT (CAR X)) V (NOT (NUMBERP (CDR X)))) (EQUAL (CAR V) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CAR, and EQUAL, to: T. Case 33.4.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS (CONS D Z1) X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS (CONS D Z1) X1))))) (NUMBERP W) X (NOT (CAR X)) (NOT (NUMBERP (CDR X)))) (EQUAL D 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CAR, and EQUAL, to: T. Case 33.4.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NUMBERP W) (NOT D) (CADR V) (NOT (NUMBERP C))) (EQUAL (CAADR V) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, CONS, EQUAL, MEMBER, and CAR, to: T. Case 33.4.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 X))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 X)))) (NUMBERP W) (NOT D) (CAR X) (NOT (NUMBERP C))) (EQUAL (CAAR X) 0)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 33.4.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS V Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS V Z1))))) (NUMBERP W) (NOT D) V (NOT (NUMBERP C))) (EQUAL (CAR V) 0)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 33.4.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS (CONS X V1) Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS (CONS X V1) Z1))))) (NUMBERP W) (NOT D) (NOT (NUMBERP C))) (EQUAL X 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 33.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAAR X))). But this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS Z1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS V W1) to eliminate (CAR Z) and (CDR Z), Z by (CONS X1 C) to eliminate (CDR Z) and (CAR Z), C by (CONS Z Z1) to eliminate (CAR C) and (CDR C), and Z by (CONS C V1) to eliminate (CAR Z) and (CDR Z). This produces the following nine new formulas: Case 33.3.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). This finally simplifies, opening up the function STRIP-CARS, to: T. Case 33.3.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NUMBERP W) V (NOT (CAR V)) (CADR Z) (NOT (NUMBERP (CDR V)))) (EQUAL (CAADR Z) 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 33.3.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 C)))) D (NUMBERP W) V (NOT (CAR V)) (CAR C) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR C) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 33.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS Z Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS Z Z1))))) D (NUMBERP W) V (NOT (CAR V)) Z (NOT (NUMBERP (CDR V)))) (EQUAL (CAR Z) 0)). However this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 33.3.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS (CONS C V1) Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS (CONS C V1) Z1))))) D (NUMBERP W) V (NOT (CAR V)) (NOT (NUMBERP (CDR V)))) (EQUAL C 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 33.3.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NUMBERP W) (NOT C) (CADR Z) (NOT (NUMBERP X1))) (EQUAL (CAADR Z) 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 33.3.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) D (NUMBERP W) (NOT C) (CAR V) (NOT (NUMBERP X1))) (EQUAL (CAAR V) 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 33.3.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS Z V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS Z V1))))) D (NUMBERP W) (NOT C) Z (NOT (NUMBERP X1))) (EQUAL (CAR Z) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and opening up STRIP-CARS and EQUAL, to: T. Case 33.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1))))) D (NUMBERP W) (NOT C) (NOT (NUMBERP X1))) (EQUAL V 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 33.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAR Z))), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS and MEMBER, to the goal: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V X) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), X by (CONS V1 V) to eliminate (CDR X) and (CAR X), V by (CONS X W1) to eliminate (CAR V) and (CDR V), X by (CONS V D1) to eliminate (CAR X) and (CDR X), X by (CONS Z1 X1) to eliminate (CDR X) and (CAR X), X1 by (CONS X V1) to eliminate (CAR X1) and (CDR X1), and X by (CONS X1 W1) to eliminate (CAR X) and (CDR X). This generates nine new formulas: Case 33.2.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 33.2.8. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V X))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (NOT (CAR V)) (CADR X) (NOT (NUMBERP (CDR V)))) (EQUAL (CAADR X) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 33.2.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (NOT (CAR V)) (CAR X1) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR X1) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 33.2.6. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS X V1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS X V1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (NOT (CAR V)) X (NOT (NUMBERP (CDR V)))) (EQUAL (CAR X) 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 33.2.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS (CONS X1 W1) V1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (NOT (CAR V)) (NOT (NUMBERP (CDR V)))) (EQUAL X1 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 33.2.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) X))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1) (CADR X) (NOT (NUMBERP Z1))) (EQUAL (CAADR X) 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 33.2.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1) (CAR V) (NOT (NUMBERP Z1))) (EQUAL (CAAR V) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS and EQUAL, to: T. Case 33.2.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS X W1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS X W1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1) X (NOT (NUMBERP Z1))) (EQUAL (CAR X) 0)). This finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 33.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1) (NOT (NUMBERP Z1))) (EQUAL V 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS and EQUAL, to: T. Case 33.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) X)). This further simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) X)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS V D1) to eliminate (CAR Z) and (CDR Z), Z by (CONS Z1 X1) to eliminate (CDR Z) and (CAR Z), X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1), and Z by (CONS X1 W1) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following nine new formulas: Case 33.1.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) X)). This finally simplifies, expanding the function STRIP-CARS, to: T. Case 33.1.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NUMBERP W) V (NOT (CAR V)) (CADR Z) (NOT (NUMBERP (CDR V)))) (EQUAL (CAADR Z) X)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 33.1.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) (NUMBERP W) V (NOT (CAR V)) (CAR X1) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR X1) X)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 33.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS Z V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS Z V1))))) D (NOT (MEMBER F C)) (NUMBERP W) V (NOT (CAR V)) Z (NOT (NUMBERP (CDR V)))) (EQUAL (CAR Z) X)). This finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and EQUAL, to: T. Case 33.1.5. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS (CONS X1 W1) V1))))) D (NOT (MEMBER F C)) (NUMBERP W) V (NOT (CAR V)) (NOT (NUMBERP (CDR V)))) (EQUAL X1 X)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 33.1.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (CADR Z) (NOT (NUMBERP Z1))) (EQUAL (CAADR Z) X)). This finally simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 33.1.3. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (CAR V) (NOT (NUMBERP Z1))) (EQUAL (CAAR V) X)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS and EQUAL, to: T. Case 33.1.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1))))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) Z (NOT (NUMBERP Z1))) (EQUAL (CAR Z) X)). But this finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 33.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1))))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (NOT (NUMBERP Z1))) (EQUAL V X)). But this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 32.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1))))) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2))))))), which again simplifies, rewriting with CAR-CONS, and unfolding the definition of EQUAL, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following nine new goals: Case 32.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADDR ARGS2))). However this further simplifies, rewriting with the lemma CAR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 32.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADR Z))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 32.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAAR V))). This further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and MEMBER, to: T. Case 32.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAR Z))). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 32.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) V)). This further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 32.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADR Z))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding the functions STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X1 X) to eliminate (CDR V) and (CAR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS X V1) to eliminate (CAR V) and (CDR V), V by (CONS C D) to eliminate (CDR V) and (CAR V), D by (CONS V X1) to eliminate (CAR D) and (CDR D), and V by (CONS D Z1) to eliminate (CAR V) and (CDR V). This produces the following nine new formulas: Case 32.4.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). However this finally simplifies, unfolding the functions STRIP-CARS and EQUAL, to: T. Case 32.4.8. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NUMBERP W) X (NOT (CAR X)) (CADR V) (NUMBERP (CDR X))) (EQUAL (CAADR V) 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, and EQUAL, to: T. Case 32.4.7. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C D))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C D)))) (NUMBERP W) X (NOT (CAR X)) (CAR D) (NUMBERP (CDR X))) (EQUAL (CAAR D) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 32.4.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS V X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS V X1))))) (NUMBERP W) X (NOT (CAR X)) V (NUMBERP (CDR X))) (EQUAL (CAR V) 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CAR, and EQUAL, to: T. Case 32.4.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C (CONS (CONS D Z1) X1)))) (LIST F)) (NOT (MEMBER F (CONS X (CONS C (CONS (CONS D Z1) X1))))) (NUMBERP W) X (NOT (CAR X)) (NUMBERP (CDR X))) (EQUAL D 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 32.4.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NUMBERP W) (NOT D) (CADR V) (NUMBERP C)) (EQUAL (CAADR V) 0)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, EQUAL, MEMBER, and CAR, to: T. Case 32.4.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 X))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 X)))) (NUMBERP W) (NOT D) (CAR X) (NUMBERP C)) (EQUAL (CAAR X) 0)), which finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 32.4.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS V Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS V Z1))))) (NUMBERP W) (NOT D) V (NUMBERP C)) (EQUAL (CAR V) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 32.4.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS (CONS X V1) Z1)))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS (CONS X V1) Z1))))) (NUMBERP W) (NOT D) (NUMBERP C)) (EQUAL X 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 32.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAAR X))). This further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS Z1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS V W1) to eliminate (CAR Z) and (CDR Z), Z by (CONS X1 C) to eliminate (CDR Z) and (CAR Z), C by (CONS Z Z1) to eliminate (CAR C) and (CDR C), and Z by (CONS C V1) to eliminate (CAR Z) and (CDR Z). We must thus prove nine new conjectures: Case 32.3.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, unfolding the function STRIP-CARS, to: T. Case 32.3.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NUMBERP W) V (NOT (CAR V)) (CADR Z) (NUMBERP (CDR V))) (EQUAL (CAADR Z) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 32.3.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 C)))) D (NUMBERP W) V (NOT (CAR V)) (CAR C) (NUMBERP (CDR V))) (EQUAL (CAAR C) 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 32.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS Z Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS Z Z1))))) D (NUMBERP W) V (NOT (CAR V)) Z (NUMBERP (CDR V))) (EQUAL (CAR Z) 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 32.3.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS (CONS C V1) Z1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS X1 (CONS (CONS C V1) Z1))))) D (NUMBERP W) V (NOT (CAR V)) (NUMBERP (CDR V))) (EQUAL C 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 32.3.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NUMBERP W) (NOT C) (CADR Z) (NUMBERP X1)) (EQUAL (CAADR Z) 0)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 32.3.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) D (NUMBERP W) (NOT C) (CAR V) (NUMBERP X1)) (EQUAL (CAAR V) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and expanding the functions STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 32.3.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS Z V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS Z V1))))) D (NUMBERP W) (NOT C) Z (NUMBERP X1)) (EQUAL (CAR Z) 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 32.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1)))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1))))) D (NUMBERP W) (NOT C) (NUMBERP X1)) (EQUAL V 0)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 32.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAR Z))), which further simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V X) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), X by (CONS V1 V) to eliminate (CDR X) and (CAR X), V by (CONS X W1) to eliminate (CAR V) and (CDR V), X by (CONS V D1) to eliminate (CAR X) and (CDR X), X by (CONS Z1 X1) to eliminate (CDR X) and (CAR X), X1 by (CONS X V1) to eliminate (CAR X1) and (CDR X1), and X by (CONS X1 W1) to eliminate (CAR X) and (CDR X). The result is nine new conjectures: Case 32.2.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, unfolding the definition of STRIP-CARS, to: T. Case 32.2.8. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V X))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (NOT (CAR V)) (CADR X) (NUMBERP (CDR V))) (EQUAL (CAADR X) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 32.2.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (NOT (CAR V)) (CAR X1) (NUMBERP (CDR V))) (EQUAL (CAAR X1) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and EQUAL, to: T. Case 32.2.6. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS X V1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS X V1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (NOT (CAR V)) X (NUMBERP (CDR V))) (EQUAL (CAR X) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 32.2.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS (CONS X1 W1) V1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (NOT (CAR V)) (NUMBERP (CDR V))) (EQUAL X1 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 32.2.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) X))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1) (CADR X) (NUMBERP Z1)) (EQUAL (CAADR X) 0)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 32.2.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1) (CAR V) (NUMBERP Z1)) (EQUAL (CAAR V) 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS and EQUAL, to: T. Case 32.2.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS X W1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS X W1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1) X (NUMBERP Z1)) (EQUAL (CAR X) 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 32.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1)))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1))))) D (NOT (MEMBER F C)) Z (NUMBERP W) (NOT X1) (NUMBERP Z1)) (EQUAL V 0)), which finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 32.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) X)). However this further simplifies, applying CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) X)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS V D1) to eliminate (CAR Z) and (CDR Z), Z by (CONS Z1 X1) to eliminate (CDR Z) and (CAR Z), X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1), and Z by (CONS X1 W1) to eliminate (CAR Z) and (CDR Z). We thus obtain the following nine new formulas: Case 32.1.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) X)). However this finally simplifies, unfolding the definition of STRIP-CARS, to: T. Case 32.1.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NUMBERP W) V (NOT (CAR V)) (CADR Z) (NUMBERP (CDR V))) (EQUAL (CAADR Z) X)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 32.1.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) (NUMBERP W) V (NOT (CAR V)) (CAR X1) (NUMBERP (CDR V))) (EQUAL (CAAR X1) X)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 32.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS Z V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS Z V1))))) D (NOT (MEMBER F C)) (NUMBERP W) V (NOT (CAR V)) Z (NUMBERP (CDR V))) (EQUAL (CAR Z) X)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 32.1.5. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS (CONS X1 W1) V1))))) D (NOT (MEMBER F C)) (NUMBERP W) V (NOT (CAR V)) (NUMBERP (CDR V))) (EQUAL X1 X)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 32.1.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (CADR Z) (NUMBERP Z1)) (EQUAL (CAADR Z) X)). But this finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 32.1.3. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (CAR V) (NUMBERP Z1)) (EQUAL (CAAR V) X)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 32.1.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1))))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) Z (NUMBERP Z1)) (EQUAL (CAR Z) X)). But this finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS, EQUAL, and MEMBER, to: T. Case 32.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS V D1) W1))))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (NUMBERP Z1)) (EQUAL V X)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 31.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2))))))). This again simplifies, rewriting with CAR-NLISTP, and unfolding the definitions of EQUAL and MEMBER, to: T. Case 30.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2))))))). However this again simplifies, appealing to the lemma CAR-CONS, and expanding the functions EQUAL and CAR, to the formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADDR ARGS2))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). The result is nine new formulas: Case 30.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADDR ARGS2))), which further simplifies, appealing to the lemma CAR-NLISTP, and opening up STRIP-CARS, MEMBER, and CAR, to: T. Case 30.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADR Z))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 30.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAAR V))). But this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 30.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAR Z))). However this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and MEMBER, to: T. Case 30.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NUMBERP (CDR X)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 V)), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 30.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADR Z))), which further simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, CONS, MEMBER, CAR, and EQUAL, to: T. Case 30.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAAR X))), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, CAR, and EQUAL, to: T. Case 30.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAR Z))), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS, MEMBER, and EQUAL, to: T. Case 30.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 X)). But this further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS and MEMBER, to the conjecture: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 X)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V V1) to eliminate (CAR Z) and (CDR Z), and Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z). We must thus prove five new formulas: Case 30.1.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 X)), which finally simplifies, expanding the function STRIP-CARS, to: T. Case 30.1.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NUMBERP W) (CAR V) (NOT (CAR Z))) (EQUAL 0 X)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 30.1.3. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) (NUMBERP W) (CAR V) (NOT X1)) (EQUAL 0 X)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CAR, and EQUAL, to: T. Case 30.1.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NUMBERP W) X1 (NOT (CAR Z))) (EQUAL 0 X)), which finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and expanding STRIP-CARS, to: T. Case 30.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) (NUMBERP W) X1 (NOT V)) (EQUAL 0 X)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS and CAR, to: T. Case 29.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2))))))). But this again simplifies, rewriting with CAR-CONS, and expanding EQUAL and CAR, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following nine new formulas: Case 29.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADDR ARGS2))). However this further simplifies, applying CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 29.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADR Z))). This further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 29.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAAR V))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 29.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAR Z))). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 29.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NUMBERP (CDR X)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 V)). This further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 29.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADR Z))). But this further simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding the functions STRIP-CARS, CONS, MEMBER, CAR, and EQUAL, to: T. Case 29.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAAR X))). But this further simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up STRIP-CARS, MEMBER, CAR, and EQUAL, to: T. Case 29.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAR Z))), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and EQUAL, to: T. Case 29.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 X)). This further simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to the new formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 X)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V1 V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS Z1 X1) to eliminate (CDR Z) and (CAR Z), and X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1). This produces the following seven new formulas: Case 29.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 X)). However this finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 29.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT (CAR V)) (NOT (CADR Z))) (EQUAL 0 X)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 29.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 X1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 X1)))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT (CAR V)) (NOT (CAR X1))) (EQUAL 0 X)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 29.1.4. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS Z1 (CONS Z V1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS Z1 (CONS Z V1))))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT (CAR V)) (NOT Z)) (EQUAL 0 X)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 29.1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (NOT (CADR Z))) (EQUAL 0 X)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 29.1.2. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 V))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 V)))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (NOT (CAR V))) (EQUAL 0 X)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 29.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1)))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS Z W1))))) D (NOT (MEMBER F C)) (NUMBERP W) (NOT X1) (NOT Z)) (EQUAL 0 X)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 28.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1))))) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2))))))). However this again simplifies, applying CAR-CONS, and opening up EQUAL and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADDR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following nine new goals: Case 28.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADDR ARGS2))). But this further simplifies, appealing to the lemma CAR-NLISTP, and opening up STRIP-CARS, MEMBER, and CAR, to: T. Case 28.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADR Z))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 28.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAAR V))). However this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 28.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAR Z))), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 28.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) V)). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 28.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADR Z))). This further simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, CONS, MEMBER, and CAR, to the new conjecture: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X X1) to eliminate (CAR V) and (CDR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS D C) to eliminate (CAR V) and (CDR V), and D by (CONS V X1) to eliminate (CAR D) and (CDR D). We thus obtain the following seven new goals: Case 28.4.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This finally simplifies, unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 28.4.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NUMBERP W) X (CAR X) (CAR V) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR V) 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, and EQUAL, to: T. Case 28.4.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS D C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS D C)))) (NUMBERP W) X (CAR X) D (NOT (NUMBERP (CDR X)))) (EQUAL (CAR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CAR, and EQUAL, to: T. Case 28.4.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS V X1) C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS (CONS V X1) C)))) (NUMBERP W) X (CAR X) (NOT (NUMBERP (CDR X)))) (EQUAL V 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 28.4.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NUMBERP W) D (CAR V) (NOT (NUMBERP C))) (EQUAL (CAAR V) 0)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and CAR, to: T. Case 28.4.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X X1)))) (NUMBERP W) D X (NOT (NUMBERP C))) (EQUAL (CAR X) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up STRIP-CARS and CAR, to: T. Case 28.4.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS (CONS V Z1) X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS (CONS V Z1) X1)))) (NUMBERP W) D (NOT (NUMBERP C))) (EQUAL V 0)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and CAR, to: T. Case 28.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAAR X))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and CAR, to the new conjecture: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS V Z1) to eliminate (CAR Z) and (CDR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z), and C by (CONS Z Z1) to eliminate (CAR C) and (CDR C). We thus obtain the following seven new formulas: Case 28.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This finally simplifies, opening up STRIP-CARS, to: T. Case 28.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NUMBERP W) V (CAR V) (CAR Z) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR Z) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 28.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS C X1)))) D (NUMBERP W) V (CAR V) C (NOT (NUMBERP (CDR V)))) (EQUAL (CAR C) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 28.3.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS Z Z1) X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS (CONS Z Z1) X1)))) D (NUMBERP W) V (CAR V) (NOT (NUMBERP (CDR V)))) (EQUAL Z 0)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 28.3.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NUMBERP W) C (CAR Z) (NOT (NUMBERP X1))) (EQUAL (CAAR Z) 0)), which finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 28.3.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) D (NUMBERP W) C V (NOT (NUMBERP X1))) (EQUAL (CAR V) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS, to: T. Case 28.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS Z V1) Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS Z V1) Z1)))) D (NUMBERP W) C (NOT (NUMBERP X1))) (EQUAL Z 0)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 28.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAR Z))). But this further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V X) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), X by (CONS V V1) to eliminate (CAR X) and (CDR X), V by (CONS X W1) to eliminate (CAR V) and (CDR V), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), and X1 by (CONS X V1) to eliminate (CAR X1) and (CDR X1). The result is seven new goals: Case 28.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)), which finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 28.2.6. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V X))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (CAR V) (CAR X) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR X) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 28.2.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (CAR V) X1 (NOT (NUMBERP (CDR V)))) (EQUAL (CAR X1) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 28.2.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS X V1) Z1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS (CONS X V1) Z1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (CAR V) (NOT (NUMBERP (CDR V)))) (EQUAL X 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 28.2.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) X))) D (NOT (MEMBER F C)) Z (NUMBERP W) X1 (CAR X) (NOT (NUMBERP Z1))) (EQUAL (CAAR X) 0)). This finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the definition of STRIP-CARS, to: T. Case 28.2.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) X1 V (NOT (NUMBERP Z1))) (EQUAL (CAR V) 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the function STRIP-CARS, to: T. Case 28.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS X W1) V1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS X W1) V1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) X1 (NOT (NUMBERP Z1))) (EQUAL X 0)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS, to: T. Case 28.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) X)). But this further simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) X)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V V1) to eliminate (CAR Z) and (CDR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), and X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1). We must thus prove seven new goals: Case 28.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) X)), which finally simplifies, opening up the function STRIP-CARS, to: T. Case 28.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NUMBERP W) V (CAR V) (CAR Z) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR Z) X)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 28.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) (NUMBERP W) V (CAR V) X1 (NOT (NUMBERP (CDR V)))) (EQUAL (CAR X1) X)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 28.1.4. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS Z V1) Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS (CONS Z V1) Z1)))) D (NOT (MEMBER F C)) (NUMBERP W) V (CAR V) (NOT (NUMBERP (CDR V)))) (EQUAL Z X)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 28.1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NUMBERP W) X1 (CAR Z) (NOT (NUMBERP Z1))) (EQUAL (CAAR Z) X)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 28.1.2. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) (NUMBERP W) X1 V (NOT (NUMBERP Z1))) (EQUAL (CAR V) X)). This finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definition of STRIP-CARS, to: T. Case 28.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS Z W1) V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS Z W1) V1)))) D (NOT (MEMBER F C)) (NUMBERP W) X1 (NOT (NUMBERP Z1))) (EQUAL Z X)). This finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 27.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1))))) (CAR (CONS (CAADDR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADDR ARGS2))))))), which again simplifies, applying the lemma CAR-CONS, and opening up EQUAL, to the goal: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADDR ARGS2))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We must thus prove nine new formulas: Case 27.9. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (NOT (CAAR ARGS2)) (CADDR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADDR ARGS2))), which further simplifies, rewriting with CAR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and CAR, to: T. Case 27.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADR Z))). But this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 27.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAAR V))). But this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 27.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAR Z))). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 27.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) V)). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and MEMBER, to: T. Case 27.4. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADR Z))). But this further simplifies, applying CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding the functions STRIP-CARS, CONS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X V) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS D C) to eliminate (CAR X) and (CDR X), V by (CONS X X1) to eliminate (CAR V) and (CDR V), X by (CONS V Z1) to eliminate (CAR X) and (CDR X), V by (CONS D C) to eliminate (CAR V) and (CDR V), and D by (CONS V X1) to eliminate (CAR D) and (CDR D). This produces the following seven new conjectures: Case 27.4.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST F)) (NOT (MEMBER F ARGS1)) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). However this finally simplifies, unfolding STRIP-CARS and EQUAL, to: T. Case 27.4.6. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X V)) (LIST F)) (NOT (MEMBER F (CONS X V))) (NUMBERP W) X (CAR X) (CAR V) (NUMBERP (CDR X))) (EQUAL (CAAR V) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, and EQUAL, to: T. Case 27.4.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS D C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS D C)))) (NUMBERP W) X (CAR X) D (NUMBERP (CDR X))) (EQUAL (CAR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CAR, and EQUAL, to: T. Case 27.4.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS V X1) C))) (LIST F)) (NOT (MEMBER F (CONS X (CONS (CONS V X1) C)))) (NUMBERP W) X (CAR X) (NUMBERP (CDR X))) (EQUAL V 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 27.4.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) V)) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) V))) (NUMBERP W) D (CAR V) (NUMBERP C)) (EQUAL (CAAR V) 0)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS and CAR, to: T. Case 27.4.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS X X1)))) (NUMBERP W) D X (NUMBERP C)) (EQUAL (CAR X) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS and CAR, to: T. Case 27.4.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS (CONS V Z1) X1))) (LIST F)) (NOT (MEMBER F (CONS (CONS D C) (CONS (CONS V Z1) X1)))) (NUMBERP W) D (NUMBERP C)) (EQUAL V 0)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding STRIP-CARS and CAR, to: T. Case 27.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAAR X))). However this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), Z by (CONS V Z1) to eliminate (CAR Z) and (CDR Z), V by (CONS Z V1) to eliminate (CAR V) and (CDR V), Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z), and C by (CONS Z Z1) to eliminate (CAR C) and (CDR C). This produces the following seven new formulas: Case 27.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (LIST F (CAR D))) (NOT (MEMBER F ARGS1)) D (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). However this finally simplifies, unfolding STRIP-CARS, to: T. Case 27.3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS V Z))) D (NUMBERP W) V (CAR V) (CAR Z) (NUMBERP (CDR V))) (EQUAL (CAAR Z) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 27.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS C X1)))) D (NUMBERP W) V (CAR V) C (NUMBERP (CDR V))) (EQUAL (CAR C) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 27.3.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS Z Z1) X1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS V (CONS (CONS Z Z1) X1)))) D (NUMBERP W) V (CAR V) (NUMBERP (CDR V))) (EQUAL Z 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 27.3.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) Z)) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) Z))) D (NUMBERP W) C (CAR Z) (NUMBERP X1)) (EQUAL (CAAR Z) 0)). This finally simplifies, applying CDR-CONS and CAR-CONS, and expanding STRIP-CARS, to: T. Case 27.3.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) D (NUMBERP W) C V (NUMBERP X1)) (EQUAL (CAR V) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the function STRIP-CARS, to: T. Case 27.3.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS Z V1) Z1))) (LIST F (CAR D))) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS Z V1) Z1)))) D (NUMBERP W) C (NUMBERP X1)) (EQUAL Z 0)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definition of STRIP-CARS, to: T. Case 27.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAR Z))). This further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V X) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), X by (CONS V V1) to eliminate (CAR X) and (CDR X), V by (CONS X W1) to eliminate (CAR V) and (CDR V), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), and X1 by (CONS X V1) to eliminate (CAR X1) and (CDR X1). We thus obtain the following seven new formulas: Case 27.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) Z (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). But this finally simplifies, unfolding STRIP-CARS, to: T. Case 27.2.6. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V X))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (CAR V) (CAR X) (NUMBERP (CDR V))) (EQUAL (CAAR X) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 27.2.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (CAR V) X1 (NUMBERP (CDR V))) (EQUAL (CAR X1) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 27.2.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS X V1) Z1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS (CONS X V1) Z1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) V (CAR V) (NUMBERP (CDR V))) (EQUAL X 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and EQUAL, to: T. Case 27.2.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) X)) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) X))) D (NOT (MEMBER F C)) Z (NUMBERP W) X1 (CAR X) (NUMBERP Z1)) (EQUAL (CAAR X) 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 27.2.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) X1 V (NUMBERP Z1)) (EQUAL (CAR V) 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS, to: T. Case 27.2.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS X W1) V1))) (CONS F (CONS (CAR D) (CONS 0 (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS X W1) V1)))) D (NOT (MEMBER F C)) Z (NUMBERP W) X1 (NUMBERP Z1)) (EQUAL X 0)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 27.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) X)). But this further simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) X)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V Z) to eliminate (CDR ARGS1) and (CAR ARGS1), V by (CONS X1 Z1) to eliminate (CAR V) and (CDR V), Z by (CONS V V1) to eliminate (CAR Z) and (CDR Z), V by (CONS Z W1) to eliminate (CAR V) and (CDR V), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), and X1 by (CONS Z V1) to eliminate (CAR X1) and (CDR X1). We must thus prove seven new conjectures: Case 27.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F ARGS1)) D (NOT (MEMBER F C)) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) X)), which finally simplifies, expanding STRIP-CARS, to: T. Case 27.1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V Z))) D (NOT (MEMBER F C)) (NUMBERP W) V (CAR V) (CAR Z) (NUMBERP (CDR V))) (EQUAL (CAAR Z) X)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 27.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS X1 Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS X1 Z1)))) D (NOT (MEMBER F C)) (NUMBERP W) V (CAR V) X1 (NUMBERP (CDR V))) (EQUAL (CAR X1) X)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 27.1.4. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS Z V1) Z1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS V (CONS (CONS Z V1) Z1)))) D (NOT (MEMBER F C)) (NUMBERP W) V (CAR V) (NUMBERP (CDR V))) (EQUAL Z X)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 27.1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) Z)) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) Z))) D (NOT (MEMBER F C)) (NUMBERP W) X1 (CAR Z) (NUMBERP Z1)) (EQUAL (CAAR Z) X)). This finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 27.1.2. (IMPLIES (AND (NOT (LISTP V)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V V1)))) D (NOT (MEMBER F C)) (NUMBERP W) X1 V (NUMBERP Z1)) (EQUAL (CAR V) X)). This finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up STRIP-CARS, to: T. Case 27.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS Z W1) V1))) (CONS F (CONS (CAR D) (CONS X (STRIP-CARS C))))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS Z W1) V1)))) D (NOT (MEMBER F C)) (NUMBERP W) X1 (NUMBERP Z1)) (EQUAL Z X)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up STRIP-CARS, to: T. Case 26.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1))))) (CAR F))). But this again simplifies, rewriting with CAR-NLISTP, and expanding the definitions of EQUAL and MEMBER, to: T. Case 25.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1))))) (CAR F))). But this again simplifies, applying CAR-CONS, and unfolding EQUAL, PLUS, and CAR, to the new formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This produces the following five new conjectures: Case 25.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). However this further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 25.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (CAR X) (NOT (CAR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CONS, and MEMBER, to: T. Case 25.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X) (NOT V) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 25.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (NOT (CAR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). But this further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 25.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V (NOT X) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 24.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1))))) (CAR F))), which again simplifies, rewriting with the lemma CAR-CONS, and expanding the definitions of EQUAL, PLUS, and CAR, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We must thus prove seven new formulas: Case 24.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, applying CAR-NLISTP, and unfolding STRIP-CARS, MEMBER, and CAR, to: T. Case 24.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X)) (NOT (CADR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). But this further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 24.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X)) (NOT (CAR V)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 24.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X)) (NOT Z) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 24.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (NOT (CADR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). But this further simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and expanding the functions STRIP-CARS, CONS, MEMBER, and CAR, to: T. Case 24.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (NOT (CAR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 24.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) (NOT Z) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 23.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1))))) (CAR F))). But this again simplifies, applying CAR-NLISTP, and unfolding the functions EQUAL and MEMBER, to: T. Case 22.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1))))) (CAR F))). However this again simplifies, rewriting with the lemma CAR-CONS, and opening up the definitions of EQUAL and CAR, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We must thus prove five new conjectures: Case 22.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, applying the lemmas CAR-NLISTP and CDR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and CAR, to: T. Case 22.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (CAR X) (NOT (CAR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 22.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X) (NOT V) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). This further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 22.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (NOT (CAR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS and MEMBER, to: T. Case 22.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V (NOT X) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 21.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1))))) (CAR F))). However this again simplifies, rewriting with CAR-CONS, and unfolding the definitions of EQUAL and CAR, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We would thus like to prove the following seven new conjectures: Case 21.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). This further simplifies, applying CAR-NLISTP, and expanding STRIP-CARS, MEMBER, and CAR, to: T. Case 21.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X)) (NOT (CADR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). But this further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, and MEMBER, to: T. Case 21.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X)) (NOT (CAR V)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 21.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X)) (NOT Z) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 21.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (NOT (CADR Z)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, CONS, MEMBER, and CAR, to: T. Case 21.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (NOT (CAR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 21.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) (NOT Z) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 20.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1))))) (CAR F))), which again simplifies, rewriting with CAR-NLISTP, and unfolding the definitions of EQUAL and MEMBER, to: T. Case 19.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1))))) (CAR F))). This again simplifies, applying CAR-CONS, and opening up EQUAL, PLUS, and CAR, to the new conjecture: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). This produces the following five new formulas: Case 19.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, applying the lemmas CAR-NLISTP and CDR-NLISTP, and unfolding STRIP-CARS, MEMBER, and CAR, to: T. Case 19.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (CAR X) (NOT (CAR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 19.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X) (NOT V) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). But this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 19.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (NOT (CAR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). However this further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 19.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V (NOT X) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)), which further simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 18.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1))))) (CAR F))), which again simplifies, applying CAR-CONS, and opening up EQUAL, PLUS, and CAR, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This produces the following seven new formulas: Case 18.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, applying CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and CAR, to: T. Case 18.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X)) (NOT (CADR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 18.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X)) (NOT (CAR V)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS and MEMBER, to: T. Case 18.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X)) (NOT Z) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 18.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (NOT (CADR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, CONS, MEMBER, and CAR, to: T. Case 18.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (NOT (CAR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). However this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS and MEMBER, to: T. Case 18.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) (NOT Z) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). But this further simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 17.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1))))) (CAR F))). This again simplifies, rewriting with CAR-NLISTP, and opening up the definitions of EQUAL and MEMBER, to: T. Case 16.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1))))) (CAR F))). This again simplifies, applying CAR-CONS, and opening up EQUAL and CAR, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), and Z by (CONS V W) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following five new goals: Case 16.5. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAAR ARGS2) (NOT (CADR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, applying the lemmas CAR-NLISTP and CDR-NLISTP, and expanding STRIP-CARS, MEMBER, and CAR, to: T. Case 16.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (CAR X) (NOT (CAR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)), which further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and MEMBER, to: T. Case 16.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) (CAR X) (NOT V) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 16.2. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (NOT (CAR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 16.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V (NOT X) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, applying CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 15.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1))))) (CAR F))). But this again simplifies, applying CAR-CONS, and opening up the definitions of EQUAL and CAR, to the new goal: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We would thus like to prove the following seven new formulas: Case 15.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (NOT (CAAR ARGS2)) (NOT (CADDR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). However this further simplifies, applying CAR-NLISTP, and expanding STRIP-CARS, MEMBER, and CAR, to: T. Case 15.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) (NOT (CAR X)) (NOT (CADR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). However this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, and MEMBER, to: T. Case 15.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W V)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W V)))) (NOT (CAR X)) (NOT (CAR V)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS and MEMBER, to: T. Case 15.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS W (CONS Z D))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) (NOT (CAR X)) (NOT Z) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). But this further simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 15.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (NOT (CADR Z)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, rewriting with CDR-CONS, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, MEMBER, and CAR, to: T. Case 15.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D X)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (NOT (CAR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). This further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 15.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) (NOT Z) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)), which further simplifies, applying CDR-CONS and CAR-CONS, and expanding STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 14.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1))))) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2))))))). However this again simplifies, rewriting with CAR-CONS, and expanding the definitions of EQUAL and PLUS, to the new conjecture: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We thus obtain the following seven new formulas: Case 14.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADR ARGS2))). This further simplifies, appealing to the lemmas CAR-NLISTP and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 14.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAAR Z))), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, CONS, MEMBER, and NUMBERP, to: T. Case 14.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAR V))). This further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 14.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). However this further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 14.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAAR Z))). This further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS Z1 X) to eliminate (CDR D) and (CAR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS X W1) to eliminate (CAR D) and (CDR D), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), C by (CONS D Z1) to eliminate (CAR C) and (CDR C), and D by (CONS C V1) to eliminate (CAR D) and (CDR D). We would thus like to prove the following nine new conjectures: Case 14.3.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). But this finally simplifies, opening up the function STRIP-CARS, to: T. Case 14.3.8. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NOT (NUMBERP W)) X (NOT (CAR X)) (CADR D) (NOT (NUMBERP (CDR X)))) (EQUAL (CAADR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CONS, CAR, EQUAL, and MEMBER, to: T. Case 14.3.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 C))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 C)))) V (NOT (NUMBERP W)) X (NOT (CAR X)) (CAR C) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR C) 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 14.3.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS D Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS D Z1))))) V (NOT (NUMBERP W)) X (NOT (CAR X)) D (NOT (NUMBERP (CDR X)))) (EQUAL (CAR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definition of STRIP-CARS, to: T. Case 14.3.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS (CONS C V1) Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS (CONS C V1) Z1))))) V (NOT (NUMBERP W)) X (NOT (CAR X)) (NOT (NUMBERP (CDR X)))) (EQUAL C 0)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definition of STRIP-CARS, to: T. Case 14.3.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NOT (NUMBERP W)) (NOT C) (CADR D) (NOT (NUMBERP X1))) (EQUAL (CAADR D) 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS, CONS, and CAR, to: T. Case 14.3.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 X))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 X)))) V (NOT (NUMBERP W)) (NOT C) (CAR X) (NOT (NUMBERP X1))) (EQUAL (CAAR X) 0)). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 14.3.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) V (NOT (NUMBERP W)) (NOT C) D (NOT (NUMBERP X1))) (EQUAL (CAR D) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definition of STRIP-CARS, to: T. Case 14.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS X W1) V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS X W1) V1))))) V (NOT (NUMBERP W)) (NOT C) (NOT (NUMBERP X1))) (EQUAL X 0)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 14.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAR X))). But this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS Z C) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), C by (CONS V1 Z) to eliminate (CDR C) and (CAR C), Z by (CONS C W1) to eliminate (CAR Z) and (CDR Z), C by (CONS Z D1) to eliminate (CAR C) and (CDR C), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1), and C by (CONS X1 W1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following nine new conjectures: Case 14.2.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). But this finally simplifies, expanding the function STRIP-CARS, to: T. Case 14.2.8. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z C))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (NOT (CAR Z)) (CADR C) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAADR C) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 14.2.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 X1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 X1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (NOT (CAR Z)) (CAR X1) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAAR X1) 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 14.2.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 (CONS C V1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (NOT (CAR Z)) C (NOT (NUMBERP (CDR Z)))) (EQUAL (CAR C) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS and MEMBER, to: T. Case 14.2.5. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 (CONS (CONS X1 W1) V1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (NOT (CAR Z)) (NOT (NUMBERP (CDR Z)))) (EQUAL X1 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS and MEMBER, to: T. Case 14.2.4. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1) (CADR C) (NOT (NUMBERP Z1))) (EQUAL (CAADR C) 0)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CONS, and CAR, to: T. Case 14.2.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 Z))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 Z)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1) (CAR Z) (NOT (NUMBERP Z1))) (EQUAL (CAAR Z) 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 14.2.2. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1) C (NOT (NUMBERP Z1))) (EQUAL (CAR C) 0)), which finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the function STRIP-CARS, to: T. Case 14.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS Z D1) W1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS Z D1) W1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1) (NOT (NUMBERP Z1))) (EQUAL Z 0)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 14.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). However this further simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to the new formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS V1 X) to eliminate (CDR C) and (CAR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS X D1) to eliminate (CAR C) and (CDR C), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1), and C by (CONS X1 W1) to eliminate (CAR C) and (CDR C). This produces the following nine new goals: Case 14.1.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). However this finally simplifies, unfolding the definition of STRIP-CARS, to: T. Case 14.1.8. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (NOT (CAR X)) (CADR C) (NOT (NUMBERP (CDR X)))) (EQUAL (CAADR C) Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CONS, and CAR, to: T. Case 14.1.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 X1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 X1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (NOT (CAR X)) (CAR X1) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR X1) Z)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 14.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS C V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (NOT (CAR X)) C (NOT (NUMBERP (CDR X)))) (EQUAL (CAR C) Z)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS and MEMBER, to: T. Case 14.1.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS (CONS X1 W1) V1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (NOT (CAR X)) (NOT (NUMBERP (CDR X)))) (EQUAL X1 Z)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 14.1.4. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (CADR C) (NOT (NUMBERP Z1))) (EQUAL (CAADR C) Z)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and CAR, to: T. Case 14.1.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 X))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 X)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (CAR X) (NOT (NUMBERP Z1))) (EQUAL (CAAR X) Z)). This finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 14.1.2. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) C (NOT (NUMBERP Z1))) (EQUAL (CAR C) Z)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definition of STRIP-CARS, to: T. Case 14.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS X D1) W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS X D1) W1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (NOT (NUMBERP Z1))) (EQUAL X Z)). But this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up the definition of STRIP-CARS, to: T. Case 13.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1))))) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2))))))), which again simplifies, applying CAR-CONS, and unfolding EQUAL and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We thus obtain the following seven new goals: Case 13.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADR ARGS2))). This further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 13.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAAR Z))). This further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, CONS, MEMBER, and NUMBERP, to: T. Case 13.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAR V))), which further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 13.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 13.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAAR Z))). However this further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS Z1 X) to eliminate (CDR D) and (CAR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS X W1) to eliminate (CAR D) and (CDR D), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), C by (CONS D Z1) to eliminate (CAR C) and (CDR C), and D by (CONS C V1) to eliminate (CAR D) and (CDR D). The result is nine new conjectures: Case 13.3.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, opening up the function STRIP-CARS, to: T. Case 13.3.8. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NOT (NUMBERP W)) X (NOT (CAR X)) (CADR D) (NUMBERP (CDR X))) (EQUAL (CAADR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, CONS, CAR, EQUAL, and MEMBER, to: T. Case 13.3.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 C))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 C)))) V (NOT (NUMBERP W)) X (NOT (CAR X)) (CAR C) (NUMBERP (CDR X))) (EQUAL (CAAR C) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the function STRIP-CARS, to: T. Case 13.3.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS D Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS D Z1))))) V (NOT (NUMBERP W)) X (NOT (CAR X)) D (NUMBERP (CDR X))) (EQUAL (CAR D) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, to: T. Case 13.3.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS (CONS C V1) Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS (CONS C V1) Z1))))) V (NOT (NUMBERP W)) X (NOT (CAR X)) (NUMBERP (CDR X))) (EQUAL C 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the function STRIP-CARS, to: T. Case 13.3.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NOT (NUMBERP W)) (NOT C) (CADR D) (NUMBERP X1)) (EQUAL (CAADR D) 0)). But this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 13.3.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 X))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 X)))) V (NOT (NUMBERP W)) (NOT C) (CAR X) (NUMBERP X1)) (EQUAL (CAAR X) 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 13.3.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) V (NOT (NUMBERP W)) (NOT C) D (NUMBERP X1)) (EQUAL (CAR D) 0)). This finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS, to: T. Case 13.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS X W1) V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS X W1) V1))))) V (NOT (NUMBERP W)) (NOT C) (NUMBERP X1)) (EQUAL X 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 13.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAR X))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS and MEMBER, to the new formula: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS Z C) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), C by (CONS V1 Z) to eliminate (CDR C) and (CAR C), Z by (CONS C W1) to eliminate (CAR Z) and (CDR Z), C by (CONS Z D1) to eliminate (CAR C) and (CDR C), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1), and C by (CONS X1 W1) to eliminate (CAR C) and (CDR C). This produces the following nine new formulas: Case 13.2.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). This finally simplifies, unfolding the function STRIP-CARS, to: T. Case 13.2.8. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z C))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (NOT (CAR Z)) (CADR C) (NUMBERP (CDR Z))) (EQUAL (CAADR C) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CONS, and CAR, to: T. Case 13.2.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 X1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 X1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (NOT (CAR Z)) (CAR X1) (NUMBERP (CDR Z))) (EQUAL (CAAR X1) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 13.2.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 (CONS C V1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (NOT (CAR Z)) C (NUMBERP (CDR Z))) (EQUAL (CAR C) 0)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 13.2.5. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 (CONS (CONS X1 W1) V1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (NOT (CAR Z)) (NUMBERP (CDR Z))) (EQUAL X1 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 13.2.4. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1) (CADR C) (NUMBERP Z1)) (EQUAL (CAADR C) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and CAR, to: T. Case 13.2.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 Z))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 Z)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1) (CAR Z) (NUMBERP Z1)) (EQUAL (CAAR Z) 0)), which finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 13.2.2. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1) C (NUMBERP Z1)) (EQUAL (CAR C) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the function STRIP-CARS, to: T. Case 13.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS Z D1) W1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS Z D1) W1))))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (NOT X1) (NUMBERP Z1)) (EQUAL Z 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 13.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)). This further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to the new formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS V1 X) to eliminate (CDR C) and (CAR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS X D1) to eliminate (CAR C) and (CDR C), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1), and C by (CONS X1 W1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following nine new goals: Case 13.1.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)). However this finally simplifies, opening up the function STRIP-CARS, to: T. Case 13.1.8. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (NOT (CAR X)) (CADR C) (NUMBERP (CDR X))) (EQUAL (CAADR C) Z)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CONS, and CAR, to: T. Case 13.1.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 X1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 X1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (NOT (CAR X)) (CAR X1) (NUMBERP (CDR X))) (EQUAL (CAAR X1) Z)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 13.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS C V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (NOT (CAR X)) C (NUMBERP (CDR X))) (EQUAL (CAR C) Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 13.1.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS (CONS X1 W1) V1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (NOT (CAR X)) (NUMBERP (CDR X))) (EQUAL X1 Z)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS and MEMBER, to: T. Case 13.1.4. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (CADR C) (NUMBERP Z1)) (EQUAL (CAADR C) Z)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 13.1.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 X))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 X)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (CAR X) (NUMBERP Z1)) (EQUAL (CAAR X) Z)). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 13.1.2. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) C (NUMBERP Z1)) (EQUAL (CAR C) Z)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the function STRIP-CARS, to: T. Case 13.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS X D1) W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS X D1) W1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (NUMBERP Z1)) (EQUAL X Z)). But this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up the definition of STRIP-CARS, to: T. Case 12.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2))))))), which again simplifies, applying CAR-NLISTP, and opening up the definitions of EQUAL and MEMBER, to: T. Case 11.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2))))))). But this again simplifies, applying CAR-CONS, and expanding the definitions of EQUAL, CAR, and PLUS, to the new formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This produces the following seven new goals: Case 11.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADR ARGS2))). This further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 11.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAAR Z))). But this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, CONS, MEMBER, and NUMBERP, to: T. Case 11.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAR V))), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 11.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NOT (NUMBERP (CDR X))) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 11.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAAR Z))). This further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, CAR, and EQUAL, to: T. Case 11.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAR X))). But this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS, MEMBER, and EQUAL, to: T. Case 11.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)). However this further simplifies, applying CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS X V1) to eliminate (CAR C) and (CDR C), and C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following five new goals: Case 11.1.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)). This finally simplifies, opening up the function STRIP-CARS, to: T. Case 11.1.4. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR X) (NOT (CAR C))) (EQUAL 0 Z)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CONS, and CAR, to: T. Case 11.1.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS X1 Z1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS X1 Z1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR X) (NOT X1)) (EQUAL 0 Z)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and CAR, to: T. Case 11.1.2. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X1 (NOT (CAR C))) (EQUAL 0 Z)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 11.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS X V1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS X V1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X1 (NOT X)) (EQUAL 0 Z)). But this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and CAR, to: T. Case 10.(IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2))))))), which again simplifies, applying CAR-CONS, and opening up the functions EQUAL, CAR, and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This produces the following seven new formulas: Case 10.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADR ARGS2))). But this further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 10.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAAR Z))). However this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, CONS, MEMBER, and NUMBERP, to: T. Case 10.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAR V))), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 10.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NOT (NUMBERP (CDR X))) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 10.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAAR Z))). This further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, CAR, and EQUAL, to: T. Case 10.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAR X))), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, and EQUAL, to: T. Case 10.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)), which further simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS V1 X) to eliminate (CDR C) and (CAR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), and X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1). We thus obtain the following seven new conjectures: Case 10.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)). However this finally simplifies, opening up the function STRIP-CARS, to: T. Case 10.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT (CAR X)) (NOT (CADR C))) (EQUAL 0 Z)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CONS, and CAR, to: T. Case 10.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 X1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 X1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT (CAR X)) (NOT (CAR X1))) (EQUAL 0 Z)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 10.1.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS C V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT (CAR X)) (NOT C)) (EQUAL 0 Z)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 10.1.3. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (NOT (CADR C))) (EQUAL 0 Z)), which finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 10.1.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 X))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 X)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (NOT (CAR X))) (EQUAL 0 Z)). This finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 10.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (NOT X1) (NOT C)) (EQUAL 0 Z)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS and CAR, to: T. Case 9. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1))))) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2))))))). However this again simplifies, rewriting with CAR-CONS, and opening up EQUAL and PLUS, to the new goal: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This produces the following seven new formulas: Case 9.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADR ARGS2))). But this further simplifies, applying CAR-NLISTP and CDR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 9.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAAR Z))). This further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, MEMBER, and NUMBERP, to: T. Case 9.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAR V))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 9.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)). This further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 9.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAAR Z))), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS X Z1) to eliminate (CAR D) and (CDR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS C X1) to eliminate (CAR D) and (CDR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). We thus obtain the following seven new conjectures: Case 9.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This finally simplifies, unfolding the function STRIP-CARS, to: T. Case 9.3.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NOT (NUMBERP W)) X (CAR X) (CAR D) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR D) 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding the functions STRIP-CARS, CONS, CAR, EQUAL, MEMBER, and NUMBERP, to: T. Case 9.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS C X1)))) V (NOT (NUMBERP W)) X (CAR X) C (NOT (NUMBERP (CDR X)))) (EQUAL (CAR C) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the function STRIP-CARS, to: T. Case 9.3.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS D Z1) X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS (CONS D Z1) X1)))) V (NOT (NUMBERP W)) X (CAR X) (NOT (NUMBERP (CDR X)))) (EQUAL D 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definition of STRIP-CARS, to: T. Case 9.3.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NOT (NUMBERP W)) C (CAR D) (NOT (NUMBERP X1))) (EQUAL (CAAR D) 0)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and unfolding the functions STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 9.3.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS X Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS X Z1)))) V (NOT (NUMBERP W)) C X (NOT (NUMBERP X1))) (EQUAL (CAR X) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding the function STRIP-CARS, to: T. Case 9.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS D V1) Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS D V1) Z1)))) V (NOT (NUMBERP W)) C (NOT (NUMBERP X1))) (EQUAL D 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the definition of STRIP-CARS, to: T. Case 9.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAR X))). However this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS Z C) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), C by (CONS Z V1) to eliminate (CAR C) and (CDR C), Z by (CONS C W1) to eliminate (CAR Z) and (CDR Z), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), and X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1). We thus obtain the following seven new goals: Case 9.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). However this finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 9.2.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z C))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (CAR Z) (CAR C) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAAR C) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, CONS, and CAR, to: T. Case 9.2.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 Z1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS X1 Z1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (CAR Z) X1 (NOT (NUMBERP (CDR Z)))) (EQUAL (CAR X1) 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, EQUAL, MEMBER, and NUMBERP, to: T. Case 9.2.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS (CONS C V1) Z1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS (CONS C V1) Z1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (CAR Z) (NOT (NUMBERP (CDR Z)))) (EQUAL C 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 9.2.3. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) X1 (CAR C) (NOT (NUMBERP Z1))) (EQUAL (CAAR C) 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the function STRIP-CARS, to: T. Case 9.2.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS Z V1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS Z V1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) X1 Z (NOT (NUMBERP Z1))) (EQUAL (CAR Z) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding the definitions of STRIP-CARS, EQUAL, and MEMBER, to: T. Case 9.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS C W1) V1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS C W1) V1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) X1 (NOT (NUMBERP Z1))) (EQUAL C 0)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 9.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)), which further simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS X V1) to eliminate (CAR C) and (CDR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), and X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1). We would thus like to prove the following seven new goals: Case 9.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)). But this finally simplifies, unfolding STRIP-CARS, to: T. Case 9.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (CAR X) (CAR C) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR C) Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS, CONS, and CAR, to: T. Case 9.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS X1 Z1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS X1 Z1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (CAR X) X1 (NOT (NUMBERP (CDR X)))) (EQUAL (CAR X1) Z)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 9.1.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS C V1) Z1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS (CONS C V1) Z1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (CAR X) (NOT (NUMBERP (CDR X)))) (EQUAL C Z)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 9.1.3. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X1 (CAR C) (NOT (NUMBERP Z1))) (EQUAL (CAAR C) Z)). However this finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 9.1.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS X V1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS X V1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X1 X (NOT (NUMBERP Z1))) (EQUAL (CAR X) Z)), which finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding the functions STRIP-CARS, MEMBER, and EQUAL, to: T. Case 9.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS C W1) V1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS C W1) V1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X1 (NOT (NUMBERP Z1))) (EQUAL C Z)). But this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the function STRIP-CARS, to: T. Case 8. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1))))) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS 0 (CDADR ARGS2))))))), which again simplifies, appealing to the lemma CAR-CONS, and expanding the definitions of EQUAL and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADR ARGS2))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This generates seven new conjectures: Case 8.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NOT (NUMBERP (CDAR ARGS2))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADR ARGS2))), which further simplifies, appealing to the lemmas CAR-NLISTP and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 8.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAAR Z))), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, CONS, MEMBER, and NUMBERP, to: T. Case 8.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAR V))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 8.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NOT (NUMBERP (CDR X))) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) Z)). But this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 8.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAAR Z))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS X Z1) to eliminate (CAR D) and (CDR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS C X1) to eliminate (CAR D) and (CDR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). This produces the following seven new goals: Case 8.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). However this finally simplifies, opening up STRIP-CARS, to: T. Case 8.3.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NOT (NUMBERP W)) X (CAR X) (CAR D) (NUMBERP (CDR X))) (EQUAL (CAAR D) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding the functions STRIP-CARS, CONS, CAR, EQUAL, MEMBER, and NUMBERP, to: T. Case 8.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS C X1)))) V (NOT (NUMBERP W)) X (CAR X) C (NUMBERP (CDR X))) (EQUAL (CAR C) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definition of STRIP-CARS, to: T. Case 8.3.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS D Z1) X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS (CONS D Z1) X1)))) V (NOT (NUMBERP W)) X (CAR X) (NUMBERP (CDR X))) (EQUAL D 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definition of STRIP-CARS, to: T. Case 8.3.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NOT (NUMBERP W)) C (CAR D) (NUMBERP X1)) (EQUAL (CAAR D) 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and opening up STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 8.3.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS X Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS X Z1)))) V (NOT (NUMBERP W)) C X (NUMBERP X1)) (EQUAL (CAR X) 0)). This finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 8.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS D V1) Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS D V1) Z1)))) V (NOT (NUMBERP W)) C (NUMBERP X1)) (EQUAL D 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the definition of STRIP-CARS, to: T. Case 8.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAR X))). However this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS Z C) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), C by (CONS Z V1) to eliminate (CAR C) and (CDR C), Z by (CONS C W1) to eliminate (CAR Z) and (CDR Z), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), and X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1). We would thus like to prove the following seven new conjectures: Case 8.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). But this finally simplifies, expanding the function STRIP-CARS, to: T. Case 8.2.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z C))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (CAR Z) (CAR C) (NUMBERP (CDR Z))) (EQUAL (CAAR C) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 8.2.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 Z1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS X1 Z1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (CAR Z) X1 (NUMBERP (CDR Z))) (EQUAL (CAR X1) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding STRIP-CARS, EQUAL, MEMBER, and NUMBERP, to: T. Case 8.2.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS (CONS C V1) Z1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS (CONS C V1) Z1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) Z (CAR Z) (NUMBERP (CDR Z))) (EQUAL C 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, to: T. Case 8.2.3. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) X1 (CAR C) (NUMBERP Z1)) (EQUAL (CAAR C) 0)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 8.2.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS Z V1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS Z V1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) X1 Z (NUMBERP Z1)) (EQUAL (CAR Z) 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 8.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS C W1) V1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS C W1) V1)))) (NOT (MEMBER F D)) V X (NOT (NUMBERP W)) X1 (NUMBERP Z1)) (EQUAL C 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 8.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) Z)), which further simplifies, applying CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS X V1) to eliminate (CAR C) and (CDR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), and X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1). We thus obtain the following seven new goals: Case 8.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) Z)). This finally simplifies, expanding STRIP-CARS, to: T. Case 8.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (CAR X) (CAR C) (NUMBERP (CDR X))) (EQUAL (CAAR C) Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS, CONS, and CAR, to: T. Case 8.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS X1 Z1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS X1 Z1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (CAR X) X1 (NUMBERP (CDR X))) (EQUAL (CAR X1) Z)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, NUMBERP, and EQUAL, to: T. Case 8.1.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS C V1) Z1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS (CONS C V1) Z1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X (CAR X) (NUMBERP (CDR X))) (EQUAL C Z)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the function STRIP-CARS, to: T. Case 8.1.3. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X1 (CAR C) (NUMBERP Z1)) (EQUAL (CAAR C) Z)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 8.1.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS X V1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS X V1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X1 X (NUMBERP Z1)) (EQUAL (CAR X) Z)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, MEMBER, and EQUAL, to: T. Case 8.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS C W1) V1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS C W1) V1)))) (NOT (MEMBER F D)) V (NOT (NUMBERP W)) X1 (NUMBERP Z1)) (EQUAL C Z)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 7. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS 0 (CDADDR ARGS1))))) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2))))))), which again simplifies, rewriting with CAR-CONS, and unfolding the functions EQUAL and PLUS, to the new goal: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This produces the following seven new formulas: Case 7.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAADR ARGS2))). This further simplifies, applying CAR-NLISTP and CDR-NLISTP, and opening up STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X Z) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We would thus like to prove the following nine new conjectures: Case 7.7.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). But this finally simplifies, applying CAR-NLISTP, and unfolding the functions STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 7.7.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X Z)) NIL) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NOT (NUMBERP (CDR X)))) (EQUAL (CAADR Z) 0)). This finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, and EQUAL, to: T. Case 7.7.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS W V))) NIL) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR V) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 7.7.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS W (CONS Z D)))) NIL) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NOT (NUMBERP (CDR X)))) (EQUAL (CAR Z) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 7.7.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D)))) NIL) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NOT (NUMBERP (CDR X)))) (EQUAL V 0)). However this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 7.7.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) Z)) NIL) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NOT (NUMBERP W))) (EQUAL (CAADR Z) 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, and EQUAL, to: T. Case 7.7.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS D X))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NOT (NUMBERP W))) (EQUAL (CAAR X) 0)). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 7.7.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C)))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NOT (NUMBERP W))) (EQUAL (CAR Z) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definition of STRIP-CARS, to: T. Case 7.7.1. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C)))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NOT (NUMBERP W))) (EQUAL X 0)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 7.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAAR Z))). However this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, CONS, MEMBER, NUMBERP, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V W) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS D C) to eliminate (CAR V) and (CDR V), W by (CONS X1 V) to eliminate (CDR W) and (CAR W), V by (CONS W Z1) to eliminate (CAR V) and (CDR V), W by (CONS V V1) to eliminate (CAR W) and (CDR W), W by (CONS C D) to eliminate (CDR W) and (CAR W), D by (CONS W X1) to eliminate (CAR D) and (CDR D), and W by (CONS D Z1) to eliminate (CAR W) and (CDR W). We would thus like to prove the following nine new conjectures: Case 7.6.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). However this finally simplifies, opening up the functions STRIP-CARS and EQUAL, to: T. Case 7.6.8. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V W)) '(0)) (NOT (MEMBER F (CONS V W))) X V (NOT (CAR V)) (CADR W) (NOT (NUMBERP (CDR V)))) (EQUAL (CAADR W) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, EQUAL, and MEMBER, to: T. Case 7.6.7. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C D))) '(0)) (NOT (MEMBER F (CONS V (CONS C D)))) X V (NOT (CAR V)) (CAR D) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR D) 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 7.6.6. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C (CONS W X1)))) '(0)) (NOT (MEMBER F (CONS V (CONS C (CONS W X1))))) X V (NOT (CAR V)) W (NOT (NUMBERP (CDR V)))) (EQUAL (CAR W) 0)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 7.6.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C (CONS (CONS D Z1) X1)))) '(0)) (NOT (MEMBER F (CONS V (CONS C (CONS (CONS D Z1) X1))))) X V (NOT (CAR V)) (NOT (NUMBERP (CDR V)))) (EQUAL D 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 7.6.4. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) W)) '(0)) (NOT (MEMBER F (CONS (CONS D C) W))) X (NOT D) (CADR W) (NOT (NUMBERP C))) (EQUAL (CAADR W) 0)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 7.6.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 V))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 V)))) X (NOT D) (CAR V) (NOT (NUMBERP C))) (EQUAL (CAAR V) 0)). This finally simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CAR, and EQUAL, to: T. Case 7.6.2. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS W Z1)))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS W Z1))))) X (NOT D) W (NOT (NUMBERP C))) (EQUAL (CAR W) 0)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 7.6.1. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS (CONS V V1) Z1)))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS (CONS V V1) Z1))))) X (NOT D) (NOT (NUMBERP C))) (EQUAL V 0)), which finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CAR, and EQUAL, to: T. Case 7.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAR V))), which further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, MEMBER, and NUMBERP, to the conjecture: (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS Z D) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z), D by (CONS Z1 Z) to eliminate (CDR D) and (CAR D), Z by (CONS D V1) to eliminate (CAR Z) and (CDR Z), D by (CONS Z W1) to eliminate (CAR D) and (CDR D), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), C by (CONS D Z1) to eliminate (CAR C) and (CDR C), and D by (CONS C V1) to eliminate (CAR D) and (CDR D). We must thus prove nine new conjectures: Case 7.5.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 7.5.8. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z D)) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z D))) (NOT (MEMBER F W)) X V Z (NOT (CAR Z)) (CADR D) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAADR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 7.5.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 C))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS X1 C)))) (NOT (MEMBER F W)) X V Z (NOT (CAR Z)) (CAR C) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAAR C) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 7.5.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 (CONS D Z1)))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS X1 (CONS D Z1))))) (NOT (MEMBER F W)) X V Z (NOT (CAR Z)) D (NOT (NUMBERP (CDR Z)))) (EQUAL (CAR D) 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 7.5.5. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 (CONS (CONS C V1) Z1)))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS X1 (CONS (CONS C V1) Z1))))) (NOT (MEMBER F W)) X V Z (NOT (CAR Z)) (NOT (NUMBERP (CDR Z)))) (EQUAL C 0)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 7.5.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X V (NOT C) (CADR D) (NOT (NUMBERP X1))) (EQUAL (CAADR D) 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 7.5.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 Z))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 Z)))) (NOT (MEMBER F W)) X V (NOT C) (CAR Z) (NOT (NUMBERP X1))) (EQUAL (CAAR Z) 0)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 7.5.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) (NOT (MEMBER F W)) X V (NOT C) D (NOT (NUMBERP X1))) (EQUAL (CAR D) 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 7.5.1. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS Z W1) V1)))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS Z W1) V1))))) (NOT (MEMBER F W)) X V (NOT C) (NOT (NUMBERP X1))) (EQUAL Z 0)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding STRIP-CARS and EQUAL, to: T. Case 7.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). But this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, and NUMBERP, to the new formula: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V D) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), D by (CONS Z1 V) to eliminate (CDR D) and (CAR D), V by (CONS D V1) to eliminate (CAR V) and (CDR V), D by (CONS V W1) to eliminate (CAR D) and (CDR D), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), C by (CONS D Z1) to eliminate (CAR C) and (CDR C), and D by (CONS C V1) to eliminate (CAR D) and (CDR D). We thus obtain the following nine new formulas: Case 7.4.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). However this finally simplifies, expanding STRIP-CARS, to: T. Case 7.4.8. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V D))) (NOT (MEMBER F W)) X V (NOT (CAR V)) (CADR D) (NOT (NUMBERP (CDR V)))) (EQUAL (CAADR D) Z)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 7.4.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS X1 C)))) (NOT (MEMBER F W)) X V (NOT (CAR V)) (CAR C) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR C) Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 7.4.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS D Z1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS X1 (CONS D Z1))))) (NOT (MEMBER F W)) X V (NOT (CAR V)) D (NOT (NUMBERP (CDR V)))) (EQUAL (CAR D) Z)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 7.4.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS (CONS C V1) Z1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS X1 (CONS (CONS C V1) Z1))))) (NOT (MEMBER F W)) X V (NOT (CAR V)) (NOT (NUMBERP (CDR V)))) (EQUAL C Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 7.4.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X (NOT C) (CADR D) (NOT (NUMBERP X1))) (EQUAL (CAADR D) Z)). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 7.4.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) (NOT (MEMBER F W)) X (NOT C) (CAR V) (NOT (NUMBERP X1))) (EQUAL (CAAR V) Z)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 7.4.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) (NOT (MEMBER F W)) X (NOT C) D (NOT (NUMBERP X1))) (EQUAL (CAR D) Z)). This finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS and EQUAL, to: T. Case 7.4.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1))))) (NOT (MEMBER F W)) X (NOT C) (NOT (NUMBERP X1))) (EQUAL V Z)), which finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 7.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAAR Z))), which further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and CAR, to the new goal: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS Z1 X) to eliminate (CDR D) and (CAR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS X W1) to eliminate (CAR D) and (CDR D), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), C by (CONS D Z1) to eliminate (CAR C) and (CDR C), and D by (CONS C V1) to eliminate (CAR D) and (CDR D). We would thus like to prove the following nine new goals: Case 7.3.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). This finally simplifies, unfolding the definition of STRIP-CARS, to: T. Case 7.3.8. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NUMBERP W) X (NOT (CAR X)) (CADR D) (NOT (NUMBERP (CDR X)))) (EQUAL (CAADR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, CONS, CAR, EQUAL, and MEMBER, to: T. Case 7.3.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 C))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 C)))) V (NUMBERP W) X (NOT (CAR X)) (CAR C) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR C) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the definition of STRIP-CARS, to: T. Case 7.3.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS D Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS D Z1))))) V (NUMBERP W) X (NOT (CAR X)) D (NOT (NUMBERP (CDR X)))) (EQUAL (CAR D) 0)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definition of STRIP-CARS, to: T. Case 7.3.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS (CONS C V1) Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS (CONS C V1) Z1))))) V (NUMBERP W) X (NOT (CAR X)) (NOT (NUMBERP (CDR X)))) (EQUAL C 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 7.3.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NUMBERP W) (NOT C) (CADR D) (NOT (NUMBERP X1))) (EQUAL (CAADR D) 0)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS, CONS, and CAR, to: T. Case 7.3.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 X))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 X)))) V (NUMBERP W) (NOT C) (CAR X) (NOT (NUMBERP X1))) (EQUAL (CAAR X) 0)). This finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 7.3.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) V (NUMBERP W) (NOT C) D (NOT (NUMBERP X1))) (EQUAL (CAR D) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definition of STRIP-CARS, to: T. Case 7.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS X W1) V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS X W1) V1))))) V (NUMBERP W) (NOT C) (NOT (NUMBERP X1))) (EQUAL X 0)). This finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 7.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) (CAR X))), which further simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS Z C) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), C by (CONS V1 Z) to eliminate (CDR C) and (CAR C), Z by (CONS C W1) to eliminate (CAR Z) and (CDR Z), C by (CONS Z D1) to eliminate (CAR C) and (CDR C), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1), and C by (CONS X1 W1) to eliminate (CAR C) and (CDR C). This generates nine new conjectures: Case 7.2.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 7.2.8. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z C))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (NOT (CAR Z)) (CADR C) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAADR C) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 7.2.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 X1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 X1)))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (NOT (CAR Z)) (CAR X1) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAAR X1) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS and MEMBER, to: T. Case 7.2.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 (CONS C V1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (NOT (CAR Z)) C (NOT (NUMBERP (CDR Z)))) (EQUAL (CAR C) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 7.2.5. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 (CONS (CONS X1 W1) V1))))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (NOT (CAR Z)) (NOT (NUMBERP (CDR Z)))) (EQUAL X1 0)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 7.2.4. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1) (CADR C) (NOT (NUMBERP Z1))) (EQUAL (CAADR C) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS, CONS, and CAR, to: T. Case 7.2.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 Z))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 Z)))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1) (CAR Z) (NOT (NUMBERP Z1))) (EQUAL (CAAR Z) 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the definition of STRIP-CARS, to: T. Case 7.2.2. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1) C (NOT (NUMBERP Z1))) (EQUAL (CAR C) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definition of STRIP-CARS, to: T. Case 7.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS Z D1) W1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS Z D1) W1))))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1) (NOT (NUMBERP Z1))) (EQUAL Z 0)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 7.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). But this further simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS V1 X) to eliminate (CDR C) and (CAR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS X D1) to eliminate (CAR C) and (CDR C), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1), and C by (CONS X1 W1) to eliminate (CAR C) and (CDR C). The result is nine new goals: Case 7.1.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADDR ARGS1) Z)), which finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 7.1.8. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NUMBERP W) X (NOT (CAR X)) (CADR C) (NOT (NUMBERP (CDR X)))) (EQUAL (CAADR C) Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS, CONS, and CAR, to: T. Case 7.1.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 X1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 X1)))) (NOT (MEMBER F D)) V (NUMBERP W) X (NOT (CAR X)) (CAR X1) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR X1) Z)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 7.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS C V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V (NUMBERP W) X (NOT (CAR X)) C (NOT (NUMBERP (CDR X)))) (EQUAL (CAR C) Z)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and MEMBER, to: T. Case 7.1.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS (CONS X1 W1) V1))))) (NOT (MEMBER F D)) V (NUMBERP W) X (NOT (CAR X)) (NOT (NUMBERP (CDR X)))) (EQUAL X1 Z)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 7.1.4. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (CADR C) (NOT (NUMBERP Z1))) (EQUAL (CAADR C) Z)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CONS, and CAR, to: T. Case 7.1.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 X))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 X)))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (CAR X) (NOT (NUMBERP Z1))) (EQUAL (CAAR X) Z)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definition of STRIP-CARS, to: T. Case 7.1.2. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) C (NOT (NUMBERP Z1))) (EQUAL (CAR C) Z)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definition of STRIP-CARS, to: T. Case 7.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS X D1) W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS X D1) W1))))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (NOT (NUMBERP Z1))) (EQUAL X Z)), which finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 6. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADDR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADDR ARGS1))))) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2))))))). This again simplifies, applying the lemma CAR-CONS, and opening up the function EQUAL, to the goal: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADR ARGS2))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We must thus prove seven new formulas: Case 6.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAADR ARGS2))), which further simplifies, applying CAR-NLISTP and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X Z) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS D X) to eliminate (CDR Z) and (CAR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS X X1) to eliminate (CAR Z) and (CDR Z), Z by (CONS W V) to eliminate (CDR Z) and (CAR Z), V by (CONS Z D) to eliminate (CAR V) and (CDR V), and Z by (CONS V C) to eliminate (CAR Z) and (CDR Z). We thus obtain the following nine new goals: Case 6.7.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). However this finally simplifies, applying CAR-NLISTP, and opening up the functions STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 6.7.8. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X Z)) NIL) (NOT (MEMBER F (CONS X Z))) X (NOT (CAR X)) (CADR Z) (NUMBERP (CDR X))) (EQUAL (CAADR Z) 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 6.7.7. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS W V))) NIL) (NOT (MEMBER F (CONS X (CONS W V)))) X (NOT (CAR X)) (CAR V) (NUMBERP (CDR X))) (EQUAL (CAAR V) 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 6.7.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS W (CONS Z D)))) NIL) (NOT (MEMBER F (CONS X (CONS W (CONS Z D))))) X (NOT (CAR X)) Z (NUMBERP (CDR X))) (EQUAL (CAR Z) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the function STRIP-CARS, to: T. Case 6.7.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS W (CONS (CONS V C) D)))) NIL) (NOT (MEMBER F (CONS X (CONS W (CONS (CONS V C) D))))) X (NOT (CAR X)) (NUMBERP (CDR X))) (EQUAL V 0)). But this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 6.7.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) Z)) NIL) (NOT (MEMBER F (CONS (CONS V W) Z))) (NOT V) (CADR Z) (NUMBERP W)) (EQUAL (CAADR Z) 0)). But this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 6.7.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS D X))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS D X)))) (NOT V) (CAR X) (NUMBERP W)) (EQUAL (CAAR X) 0)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 6.7.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS D (CONS Z C)))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS Z C))))) (NOT V) Z (NUMBERP W)) (EQUAL (CAR Z) 0)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS, to: T. Case 6.7.1. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS D (CONS (CONS X X1) C)))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS D (CONS (CONS X X1) C))))) (NOT V) (NUMBERP W)) (EQUAL X 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 6.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAAR Z))). However this further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, CONS, MEMBER, NUMBERP, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V W) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS D C) to eliminate (CAR V) and (CDR V), W by (CONS X1 V) to eliminate (CDR W) and (CAR W), V by (CONS W Z1) to eliminate (CAR V) and (CDR V), W by (CONS V V1) to eliminate (CAR W) and (CDR W), W by (CONS C D) to eliminate (CDR W) and (CAR W), D by (CONS W X1) to eliminate (CAR D) and (CDR D), and W by (CONS D Z1) to eliminate (CAR W) and (CDR W). We must thus prove nine new formulas: Case 6.6.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 6.6.8. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V W)) '(0)) (NOT (MEMBER F (CONS V W))) X V (NOT (CAR V)) (CADR W) (NUMBERP (CDR V))) (EQUAL (CAADR W) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, EQUAL, and MEMBER, to: T. Case 6.6.7. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C D))) '(0)) (NOT (MEMBER F (CONS V (CONS C D)))) X V (NOT (CAR V)) (CAR D) (NUMBERP (CDR V))) (EQUAL (CAAR D) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 6.6.6. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C (CONS W X1)))) '(0)) (NOT (MEMBER F (CONS V (CONS C (CONS W X1))))) X V (NOT (CAR V)) W (NUMBERP (CDR V))) (EQUAL (CAR W) 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 6.6.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C (CONS (CONS D Z1) X1)))) '(0)) (NOT (MEMBER F (CONS V (CONS C (CONS (CONS D Z1) X1))))) X V (NOT (CAR V)) (NUMBERP (CDR V))) (EQUAL D 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the definitions of STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 6.6.4. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) W)) '(0)) (NOT (MEMBER F (CONS (CONS D C) W))) X (NOT D) (CADR W) (NUMBERP C)) (EQUAL (CAADR W) 0)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 6.6.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 V))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 V)))) X (NOT D) (CAR V) (NUMBERP C)) (EQUAL (CAAR V) 0)). But this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CAR, and EQUAL, to: T. Case 6.6.2. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS W Z1)))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS W Z1))))) X (NOT D) W (NUMBERP C)) (EQUAL (CAR W) 0)), which finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS, CAR, and EQUAL, to: T. Case 6.6.1. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS X1 (CONS (CONS V V1) Z1)))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS X1 (CONS (CONS V V1) Z1))))) X (NOT D) (NUMBERP C)) (EQUAL V 0)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS, CAR, and EQUAL, to: T. Case 6.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAR V))). But this further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the functions STRIP-CARS, MEMBER, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS Z D) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z), D by (CONS Z1 Z) to eliminate (CDR D) and (CAR D), Z by (CONS D V1) to eliminate (CAR Z) and (CDR Z), D by (CONS Z W1) to eliminate (CAR D) and (CDR D), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), C by (CONS D Z1) to eliminate (CAR C) and (CDR C), and D by (CONS C V1) to eliminate (CAR D) and (CDR D). This generates nine new conjectures: Case 6.5.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 6.5.8. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z D)) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z D))) (NOT (MEMBER F W)) X V Z (NOT (CAR Z)) (CADR D) (NUMBERP (CDR Z))) (EQUAL (CAADR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 6.5.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 C))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS X1 C)))) (NOT (MEMBER F W)) X V Z (NOT (CAR Z)) (CAR C) (NUMBERP (CDR Z))) (EQUAL (CAAR C) 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, EQUAL, and MEMBER, to: T. Case 6.5.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 (CONS D Z1)))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS X1 (CONS D Z1))))) (NOT (MEMBER F W)) X V Z (NOT (CAR Z)) D (NUMBERP (CDR Z))) (EQUAL (CAR D) 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 6.5.5. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 (CONS (CONS C V1) Z1)))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS X1 (CONS (CONS C V1) Z1))))) (NOT (MEMBER F W)) X V Z (NOT (CAR Z)) (NUMBERP (CDR Z))) (EQUAL C 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 6.5.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X V (NOT C) (CADR D) (NUMBERP X1)) (EQUAL (CAADR D) 0)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 6.5.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 Z))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 Z)))) (NOT (MEMBER F W)) X V (NOT C) (CAR Z) (NUMBERP X1)) (EQUAL (CAAR Z) 0)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 6.5.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) (NOT (MEMBER F W)) X V (NOT C) D (NUMBERP X1)) (EQUAL (CAR D) 0)). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 6.5.1. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS Z W1) V1)))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS Z W1) V1))))) (NOT (MEMBER F W)) X V (NOT C) (NUMBERP X1)) (EQUAL Z 0)). However this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 6.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NUMBERP (CDR X)) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V D) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), D by (CONS Z1 V) to eliminate (CDR D) and (CAR D), V by (CONS D V1) to eliminate (CAR V) and (CDR V), D by (CONS V W1) to eliminate (CAR D) and (CDR D), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), C by (CONS D Z1) to eliminate (CAR C) and (CDR C), and D by (CONS C V1) to eliminate (CAR D) and (CDR D). We must thus prove nine new goals: Case 6.4.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)), which finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 6.4.8. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V D))) (NOT (MEMBER F W)) X V (NOT (CAR V)) (CADR D) (NUMBERP (CDR V))) (EQUAL (CAADR D) Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 6.4.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS X1 C)))) (NOT (MEMBER F W)) X V (NOT (CAR V)) (CAR C) (NUMBERP (CDR V))) (EQUAL (CAAR C) Z)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS, EQUAL, and MEMBER, to: T. Case 6.4.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS D Z1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS X1 (CONS D Z1))))) (NOT (MEMBER F W)) X V (NOT (CAR V)) D (NUMBERP (CDR V))) (EQUAL (CAR D) Z)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS, EQUAL, and MEMBER, to: T. Case 6.4.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS (CONS C V1) Z1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS X1 (CONS (CONS C V1) Z1))))) (NOT (MEMBER F W)) X V (NOT (CAR V)) (NUMBERP (CDR V))) (EQUAL C Z)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 6.4.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X (NOT C) (CADR D) (NUMBERP X1)) (EQUAL (CAADR D) Z)), which finally simplifies, applying CDR-CONS and CAR-CONS, and opening up STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 6.4.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) (NOT (MEMBER F W)) X (NOT C) (CAR V) (NUMBERP X1)) (EQUAL (CAAR V) Z)). This finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS and EQUAL, to: T. Case 6.4.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) (NOT (MEMBER F W)) X (NOT C) D (NUMBERP X1)) (EQUAL (CAR D) Z)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS and EQUAL, to: T. Case 6.4.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS V W1) V1))))) (NOT (MEMBER F W)) X (NOT C) (NUMBERP X1)) (EQUAL V Z)). But this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS and EQUAL, to: T. Case 6.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAAR Z))), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding STRIP-CARS, MEMBER, and CAR, to the conjecture: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS Z1 X) to eliminate (CDR D) and (CAR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS X W1) to eliminate (CAR D) and (CDR D), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), C by (CONS D Z1) to eliminate (CAR C) and (CDR C), and D by (CONS C V1) to eliminate (CAR D) and (CDR D). We must thus prove nine new conjectures: Case 6.3.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)), which finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 6.3.8. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NUMBERP W) X (NOT (CAR X)) (CADR D) (NUMBERP (CDR X))) (EQUAL (CAADR D) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS, CONS, CAR, EQUAL, and MEMBER, to: T. Case 6.3.7. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 C))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 C)))) V (NUMBERP W) X (NOT (CAR X)) (CAR C) (NUMBERP (CDR X))) (EQUAL (CAAR C) 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 6.3.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS D Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS D Z1))))) V (NUMBERP W) X (NOT (CAR X)) D (NUMBERP (CDR X))) (EQUAL (CAR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the function STRIP-CARS, to: T. Case 6.3.5. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS X1 (CONS (CONS C V1) Z1)))) (LIST V)) (NOT (MEMBER F (CONS X (CONS X1 (CONS (CONS C V1) Z1))))) V (NUMBERP W) X (NOT (CAR X)) (NUMBERP (CDR X))) (EQUAL C 0)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, to: T. Case 6.3.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NUMBERP W) (NOT C) (CADR D) (NUMBERP X1)) (EQUAL (CAADR D) 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, and CAR, to: T. Case 6.3.3. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 X))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 X)))) V (NUMBERP W) (NOT C) (CAR X) (NUMBERP X1)) (EQUAL (CAAR X) 0)). But this finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up the definition of STRIP-CARS, to: T. Case 6.3.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) V (NUMBERP W) (NOT C) D (NUMBERP X1)) (EQUAL (CAR D) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the function STRIP-CARS, to: T. Case 6.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS (CONS X W1) V1)))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS (CONS X W1) V1))))) V (NUMBERP W) (NOT C) (NUMBERP X1)) (EQUAL X 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 6.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) (CAR X))). But this further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the functions STRIP-CARS and MEMBER, to the new formula: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS Z C) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), C by (CONS V1 Z) to eliminate (CDR C) and (CAR C), Z by (CONS C W1) to eliminate (CAR Z) and (CDR Z), C by (CONS Z D1) to eliminate (CAR C) and (CDR C), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1), and C by (CONS X1 W1) to eliminate (CAR C) and (CDR C). This produces the following nine new formulas: Case 6.2.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) 0)). This finally simplifies, expanding the function STRIP-CARS, to: T. Case 6.2.8. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z C))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (NOT (CAR Z)) (CADR C) (NUMBERP (CDR Z))) (EQUAL (CAADR C) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 6.2.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 X1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 X1)))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (NOT (CAR Z)) (CAR X1) (NUMBERP (CDR Z))) (EQUAL (CAAR X1) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and MEMBER, to: T. Case 6.2.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 (CONS C V1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (NOT (CAR Z)) C (NUMBERP (CDR Z))) (EQUAL (CAR C) 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS and MEMBER, to: T. Case 6.2.5. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS Z1 (CONS (CONS X1 W1) V1))))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (NOT (CAR Z)) (NUMBERP (CDR Z))) (EQUAL X1 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definitions of STRIP-CARS and MEMBER, to: T. Case 6.2.4. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1) (CADR C) (NUMBERP Z1)) (EQUAL (CAADR C) 0)), which finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 6.2.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 Z))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 Z)))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1) (CAR Z) (NUMBERP Z1)) (EQUAL (CAAR Z) 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 6.2.2. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1) C (NUMBERP Z1)) (EQUAL (CAR C) 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definition of STRIP-CARS, to: T. Case 6.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS Z D1) W1)))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS Z D1) W1))))) (NOT (MEMBER F D)) V X (NUMBERP W) (NOT X1) (NUMBERP Z1)) (EQUAL Z 0)), which finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 6.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)). But this further simplifies, applying CDR-CONS and CAR-CONS, and unfolding the functions STRIP-CARS and MEMBER, to the new goal: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS V1 X) to eliminate (CDR C) and (CAR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS X D1) to eliminate (CAR C) and (CDR C), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1), and C by (CONS X1 W1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following nine new formulas: Case 6.1.9. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (CAR ARGS1) (NOT (CAAR ARGS1)) (CADDR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADDR ARGS1) Z)). This finally simplifies, unfolding the function STRIP-CARS, to: T. Case 6.1.8. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NUMBERP W) X (NOT (CAR X)) (CADR C) (NUMBERP (CDR X))) (EQUAL (CAADR C) Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, CONS, and CAR, to: T. Case 6.1.7. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 X1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 X1)))) (NOT (MEMBER F D)) V (NUMBERP W) X (NOT (CAR X)) (CAR X1) (NUMBERP (CDR X))) (EQUAL (CAAR X1) Z)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS and MEMBER, to: T. Case 6.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS C V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V (NUMBERP W) X (NOT (CAR X)) C (NUMBERP (CDR X))) (EQUAL (CAR C) Z)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and MEMBER, to: T. Case 6.1.5. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS (CONS X1 W1) V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS (CONS X1 W1) V1))))) (NOT (MEMBER F D)) V (NUMBERP W) X (NOT (CAR X)) (NUMBERP (CDR X))) (EQUAL X1 Z)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS and MEMBER, to: T. Case 6.1.4. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (CADR C) (NUMBERP Z1)) (EQUAL (CAADR C) Z)). However this finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding STRIP-CARS, CONS, and CAR, to: T. Case 6.1.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 X))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 X)))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (CAR X) (NUMBERP Z1)) (EQUAL (CAAR X) Z)). But this finally simplifies, applying CDR-CONS and CAR-CONS, and expanding STRIP-CARS, to: T. Case 6.1.2. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) C (NUMBERP Z1)) (EQUAL (CAR C) Z)). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding STRIP-CARS, to: T. Case 6.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS X D1) W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS (CONS X D1) W1))))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (NUMBERP Z1)) (EQUAL X Z)). However this finally simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 5. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2))))))), which again simplifies, rewriting with CAR-NLISTP, and unfolding EQUAL and MEMBER, to: T. Case 4. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2))))))). But this again simplifies, rewriting with the lemma CAR-CONS, and opening up the definitions of EQUAL and CAR, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADR ARGS2))). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). The result is seven new goals: Case 4.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAADR ARGS2))), which further simplifies, applying CAR-NLISTP and CDR-NLISTP, and expanding STRIP-CARS, MEMBER, CAR, CDR, NUMBERP, and EQUAL, to: T. Case 4.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAAR Z))). This further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, CONS, MEMBER, NUMBERP, CAR, and EQUAL, to: T. Case 4.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAR V))). This further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up STRIP-CARS, MEMBER, NUMBERP, and EQUAL, to: T. Case 4.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NUMBERP (CDR X)) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)). But this further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and NUMBERP, to the formula: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V D) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), D by (CONS V Z1) to eliminate (CAR D) and (CDR D), and D by (CONS C X1) to eliminate (CAR D) and (CDR D). This generates five new formulas: Case 4.4.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)), which finally simplifies, opening up STRIP-CARS, to: T. Case 4.4.4. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V D))) (NOT (MEMBER F W)) X (CAR V) (NOT (CAR D))) (EQUAL 0 Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 4.4.3. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS C X1)))) (NOT (MEMBER F W)) X (CAR V) (NOT C)) (EQUAL 0 Z)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the definitions of STRIP-CARS, CAR, and EQUAL, to: T. Case 4.4.2. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X C (NOT (CAR D))) (EQUAL 0 Z)), which finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the function STRIP-CARS, to: T. Case 4.4.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) (NOT (MEMBER F W)) X C (NOT V)) (EQUAL 0 Z)). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS and CAR, to: T. Case 4.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAAR Z))). But this further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, CAR, and EQUAL, to: T. Case 4.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 (CAR X))), which further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and EQUAL, to: T. Case 4.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)), which further simplifies, rewriting with CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS X V1) to eliminate (CAR C) and (CDR C), and C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C). This produces the following five new conjectures: Case 4.1.5. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (CAAR ARGS1) (NOT (CADR ARGS1))) (EQUAL 0 Z)). But this finally simplifies, opening up the function STRIP-CARS, to: T. Case 4.1.4. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NUMBERP W) (CAR X) (NOT (CAR C))) (EQUAL 0 Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS, CONS, and CAR, to: T. Case 4.1.3. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS X1 Z1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS X1 Z1)))) (NOT (MEMBER F D)) V (NUMBERP W) (CAR X) (NOT X1)) (EQUAL 0 Z)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and CAR, to: T. Case 4.1.2. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NUMBERP W) X1 (NOT (CAR C))) (EQUAL 0 Z)). This finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, to: T. Case 4.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS X V1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS X V1)))) (NOT (MEMBER F D)) V (NUMBERP W) X1 (NOT X)) (EQUAL 0 Z)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the functions STRIP-CARS and CAR, to: T. Case 3. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL (CAR F) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2))))))). This again simplifies, rewriting with CAR-CONS, and opening up the functions EQUAL and CAR, to the new formula: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We thus obtain the following seven new conjectures: Case 3.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAADR ARGS2))). This further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, CAR, CDR, NUMBERP, and EQUAL, to: T. Case 3.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAAR Z))). But this further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, CONS, MEMBER, NUMBERP, CAR, and EQUAL, to: T. Case 3.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAR V))), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, NUMBERP, and EQUAL, to: T. Case 3.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NUMBERP (CDR X)) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)). This further simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V D) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), D by (CONS Z1 V) to eliminate (CDR D) and (CAR D), V by (CONS D V1) to eliminate (CAR V) and (CDR V), D by (CONS X1 C) to eliminate (CDR D) and (CAR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). The result is seven new formulas: Case 3.4.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)), which finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 3.4.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V D))) (NOT (MEMBER F W)) X (NOT (CAR V)) (NOT (CADR D))) (EQUAL 0 Z)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 3.4.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 C))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS X1 C)))) (NOT (MEMBER F W)) X (NOT (CAR V)) (NOT (CAR C))) (EQUAL 0 Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, EQUAL, and MEMBER, to: T. Case 3.4.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS X1 (CONS D Z1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS X1 (CONS D Z1))))) (NOT (MEMBER F W)) X (NOT (CAR V)) (NOT D)) (EQUAL 0 Z)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CAR, EQUAL, and MEMBER, to: T. Case 3.4.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X (NOT C) (NOT (CADR D))) (EQUAL 0 Z)), which finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and opening up the definitions of STRIP-CARS, CONS, CAR, and EQUAL, to: T. Case 3.4.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 V))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 V)))) (NOT (MEMBER F W)) X (NOT C) (NOT (CAR V))) (EQUAL 0 Z)), which finally simplifies, rewriting with the lemmas CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 3.4.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z1 (CONS D V1)))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z1 (CONS D V1))))) (NOT (MEMBER F W)) X (NOT C) (NOT D)) (EQUAL 0 Z)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding STRIP-CARS, CAR, and EQUAL, to: T. Case 3.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAAR Z))). But this further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, CAR, and EQUAL, to: T. Case 3.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 (CAR X))). However this further simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS, MEMBER, and EQUAL, to: T. Case 3.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)), which further simplifies, applying CDR-CONS and CAR-CONS, and expanding the functions STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS V1 X) to eliminate (CDR C) and (CAR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS Z1 X1) to eliminate (CDR C) and (CAR C), and X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1). We thus obtain the following seven new formulas: Case 3.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (NOT (CAAR ARGS1)) (NOT (CADDR ARGS1))) (EQUAL 0 Z)). This finally simplifies, opening up STRIP-CARS, to: T. Case 3.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT (CAR X)) (NOT (CADR C))) (EQUAL 0 Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, CONS, and CAR, to: T. Case 3.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 X1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 X1)))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT (CAR X)) (NOT (CAR X1))) (EQUAL 0 Z)). This finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS and MEMBER, to: T. Case 3.1.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS Z1 (CONS C V1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS Z1 (CONS C V1))))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT (CAR X)) (NOT C)) (EQUAL 0 Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS, CAR, MEMBER, and EQUAL, to: T. Case 3.1.3. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (NOT (CADR C))) (EQUAL 0 Z)). However this finally simplifies, applying CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, CONS, and CAR, to: T. Case 3.1.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 X))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 X)))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (NOT (CAR X))) (EQUAL 0 Z)). This finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and unfolding the definition of STRIP-CARS, to: T. Case 3.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS V1 (CONS C W1)))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS V1 (CONS C W1))))) (NOT (MEMBER F D)) V (NUMBERP W) (NOT X1) (NOT C)) (EQUAL 0 Z)), which finally simplifies, applying CDR-CONS and CAR-CONS, and opening up the functions STRIP-CARS and CAR, to: T. Case 2. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS 0 (CDADR ARGS1))))) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2))))))). But this again simplifies, applying CAR-CONS, and opening up EQUAL and PLUS, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This produces the following seven new conjectures: Case 2.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAADR ARGS2))). But this further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to the new formula: (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X Z) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). This produces the following seven new goals: Case 2.7.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This finally simplifies, applying the lemmas CAR-NLISTP and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, EQUAL, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 2.7.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X Z)) NIL) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR Z) 0)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, and EQUAL, to: T. Case 2.7.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS V W))) NIL) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NOT (NUMBERP (CDR X)))) (EQUAL (CAR V) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 2.7.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS Z D) W))) NIL) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NOT (NUMBERP (CDR X)))) (EQUAL Z 0)). But this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definition of STRIP-CARS, to: T. Case 2.7.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) Z)) NIL) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NOT (NUMBERP W))) (EQUAL (CAAR Z) 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and expanding the function STRIP-CARS, to: T. Case 2.7.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS X D))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NOT (NUMBERP W))) (EQUAL (CAR X) 0)). This finally simplifies, appealing to the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definition of STRIP-CARS, to: T. Case 2.7.1. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NOT (NUMBERP W))) (EQUAL Z 0)), which finally simplifies, appealing to the lemmas CDR-CONS and CAR-CONS, and opening up STRIP-CARS, to: T. Case 2.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAAR Z))), which further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, CONS, MEMBER, NUMBERP, and CAR, to the new goal: (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V W) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS D C) to eliminate (CAR V) and (CDR V), W by (CONS V X1) to eliminate (CAR W) and (CDR W), V by (CONS W Z1) to eliminate (CAR V) and (CDR V), W by (CONS D C) to eliminate (CAR W) and (CDR W), and D by (CONS W X1) to eliminate (CAR D) and (CDR D). We would thus like to prove the following seven new conjectures: Case 2.6.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). However this finally simplifies, unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 2.6.6. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V W)) '(0)) (NOT (MEMBER F (CONS V W))) X V (CAR V) (CAR W) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR W) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, CONS, EQUAL, MEMBER, and NUMBERP, to: T. Case 2.6.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS D C))) '(0)) (NOT (MEMBER F (CONS V (CONS D C)))) X V (CAR V) D (NOT (NUMBERP (CDR V)))) (EQUAL (CAR D) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 2.6.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS W X1) C))) '(0)) (NOT (MEMBER F (CONS V (CONS (CONS W X1) C)))) X V (CAR V) (NOT (NUMBERP (CDR V)))) (EQUAL W 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 2.6.3. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) W)) '(0)) (NOT (MEMBER F (CONS (CONS D C) W))) X D (CAR W) (NOT (NUMBERP C))) (EQUAL (CAAR W) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and expanding STRIP-CARS, CAR, EQUAL, and MEMBER, to: T. Case 2.6.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS V X1))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS V X1)))) X D V (NOT (NUMBERP C))) (EQUAL (CAR V) 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and opening up the functions STRIP-CARS and CAR, to: T. Case 2.6.1. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS (CONS W Z1) X1))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS (CONS W Z1) X1)))) X D (NOT (NUMBERP C))) (EQUAL W 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the functions STRIP-CARS and CAR, to: T. Case 2.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAR V))), which further simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS Z D) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z), D by (CONS Z Z1) to eliminate (CAR D) and (CDR D), Z by (CONS D V1) to eliminate (CAR Z) and (CDR Z), D by (CONS C X1) to eliminate (CAR D) and (CDR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). We must thus prove seven new conjectures: Case 2.5.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)), which finally simplifies, unfolding the function STRIP-CARS, to: T. Case 2.5.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z D)) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z D))) (NOT (MEMBER F W)) X V Z (CAR Z) (CAR D) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAAR D) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 2.5.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS C X1))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS C X1)))) (NOT (MEMBER F W)) X V Z (CAR Z) C (NOT (NUMBERP (CDR Z)))) (EQUAL (CAR C) 0)). This finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and unfolding the functions STRIP-CARS, EQUAL, MEMBER, and NUMBERP, to: T. Case 2.5.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS (CONS D Z1) X1))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS (CONS D Z1) X1)))) (NOT (MEMBER F W)) X V Z (CAR Z) (NOT (NUMBERP (CDR Z)))) (EQUAL D 0)). This finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS and EQUAL, to: T. Case 2.5.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X V C (CAR D) (NOT (NUMBERP X1))) (EQUAL (CAAR D) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 2.5.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z Z1))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z Z1)))) (NOT (MEMBER F W)) X V C Z (NOT (NUMBERP X1))) (EQUAL (CAR Z) 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS, EQUAL, and MEMBER, to: T. Case 2.5.1. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS D V1) Z1))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS D V1) Z1)))) (NOT (MEMBER F W)) X V C (NOT (NUMBERP X1))) (EQUAL D 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the definition of STRIP-CARS, to: T. Case 2.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)), which further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and opening up the functions STRIP-CARS, MEMBER, and NUMBERP, to the new conjecture: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V D) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), D by (CONS V Z1) to eliminate (CAR D) and (CDR D), V by (CONS D V1) to eliminate (CAR V) and (CDR V), D by (CONS C X1) to eliminate (CAR D) and (CDR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following seven new conjectures: Case 2.4.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)). But this finally simplifies, expanding the function STRIP-CARS, to: T. Case 2.4.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V D))) (NOT (MEMBER F W)) X V (CAR V) (CAR D) (NOT (NUMBERP (CDR V)))) (EQUAL (CAAR D) Z)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 2.4.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS C X1)))) (NOT (MEMBER F W)) X V (CAR V) C (NOT (NUMBERP (CDR V)))) (EQUAL (CAR C) Z)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding the functions STRIP-CARS, EQUAL, MEMBER, and NUMBERP, to: T. Case 2.4.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS D Z1) X1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS (CONS D Z1) X1)))) (NOT (MEMBER F W)) X V (CAR V) (NOT (NUMBERP (CDR V)))) (EQUAL D Z)). However this finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 2.4.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X C (CAR D) (NOT (NUMBERP X1))) (EQUAL (CAAR D) Z)), which finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definition of STRIP-CARS, to: T. Case 2.4.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) (NOT (MEMBER F W)) X C V (NOT (NUMBERP X1))) (EQUAL (CAR V) Z)). However this finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding STRIP-CARS, MEMBER, and EQUAL, to: T. Case 2.4.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS D V1) Z1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS D V1) Z1)))) (NOT (MEMBER F W)) X C (NOT (NUMBERP X1))) (EQUAL D Z)), which finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, to: T. Case 2.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAAR Z))), which further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the functions STRIP-CARS, MEMBER, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS X Z1) to eliminate (CAR D) and (CDR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS C X1) to eliminate (CAR D) and (CDR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following seven new conjectures: Case 2.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). This finally simplifies, unfolding the function STRIP-CARS, to: T. Case 2.3.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NUMBERP W) X (CAR X) (CAR D) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR D) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and opening up the definitions of STRIP-CARS, CONS, CAR, EQUAL, MEMBER, and NUMBERP, to: T. Case 2.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS C X1)))) V (NUMBERP W) X (CAR X) C (NOT (NUMBERP (CDR X)))) (EQUAL (CAR C) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the function STRIP-CARS, to: T. Case 2.3.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS D Z1) X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS (CONS D Z1) X1)))) V (NUMBERP W) X (CAR X) (NOT (NUMBERP (CDR X)))) (EQUAL D 0)). This finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up the definition of STRIP-CARS, to: T. Case 2.3.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NUMBERP W) C (CAR D) (NOT (NUMBERP X1))) (EQUAL (CAAR D) 0)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and expanding the definitions of STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 2.3.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS X Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS X Z1)))) V (NUMBERP W) C X (NOT (NUMBERP X1))) (EQUAL (CAR X) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 2.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS D V1) Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS D V1) Z1)))) V (NUMBERP W) C (NOT (NUMBERP X1))) (EQUAL D 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definition of STRIP-CARS, to: T. Case 2.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) (CAR X))). This further simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the definitions of STRIP-CARS and MEMBER, to: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS Z C) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), C by (CONS Z V1) to eliminate (CAR C) and (CDR C), Z by (CONS C W1) to eliminate (CAR Z) and (CDR Z), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), and X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1). This generates seven new conjectures: Case 2.2.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) 0)), which finally simplifies, expanding STRIP-CARS, to: T. Case 2.2.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z C))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (CAR Z) (CAR C) (NOT (NUMBERP (CDR Z)))) (EQUAL (CAAR C) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding the definitions of STRIP-CARS, CONS, and CAR, to: T. Case 2.2.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS X1 Z1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS X1 Z1)))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (CAR Z) X1 (NOT (NUMBERP (CDR Z)))) (EQUAL (CAR X1) 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding the definitions of STRIP-CARS, EQUAL, MEMBER, and NUMBERP, to: T. Case 2.2.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS (CONS C V1) Z1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS Z (CONS (CONS C V1) Z1)))) (NOT (MEMBER F D)) V X (NUMBERP W) Z (CAR Z) (NOT (NUMBERP (CDR Z)))) (EQUAL C 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the definition of STRIP-CARS, to: T. Case 2.2.3. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V X (NUMBERP W) X1 (CAR C) (NOT (NUMBERP Z1))) (EQUAL (CAAR C) 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definition of STRIP-CARS, to: T. Case 2.2.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS Z V1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS Z V1)))) (NOT (MEMBER F D)) V X (NUMBERP W) X1 Z (NOT (NUMBERP Z1))) (EQUAL (CAR Z) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and expanding the definitions of STRIP-CARS, EQUAL, and MEMBER, to: T. Case 2.2.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS C W1) V1))) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS C W1) V1)))) (NOT (MEMBER F D)) V X (NUMBERP W) X1 (NOT (NUMBERP Z1))) (EQUAL C 0)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the definition of STRIP-CARS, to: T. Case 2.1. (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)). This further simplifies, applying the lemmas CDR-CONS and CAR-CONS, and unfolding STRIP-CARS and MEMBER, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS X C) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS X1 Z1) to eliminate (CAR X) and (CDR X), C by (CONS X V1) to eliminate (CAR C) and (CDR C), X by (CONS C W1) to eliminate (CAR X) and (CDR X), C by (CONS X1 Z1) to eliminate (CAR C) and (CDR C), and X1 by (CONS C V1) to eliminate (CAR X1) and (CDR X1). The result is seven new formulas: Case 2.1.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NOT (NUMBERP (CDAR ARGS1)))) (EQUAL (CAADR ARGS1) Z)), which finally simplifies, expanding the function STRIP-CARS, to: T. Case 2.1.6. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X C))) (NOT (MEMBER F D)) V (NUMBERP W) X (CAR X) (CAR C) (NOT (NUMBERP (CDR X)))) (EQUAL (CAAR C) Z)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS, CONS, and CAR, to: T. Case 2.1.5. (IMPLIES (AND (NOT (LISTP X1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS X1 Z1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS X1 Z1)))) (NOT (MEMBER F D)) V (NUMBERP W) X (CAR X) X1 (NOT (NUMBERP (CDR X)))) (EQUAL (CAR X1) Z)), which finally simplifies, appealing to the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding STRIP-CARS, MEMBER, and NUMBERP, to: T. Case 2.1.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS C V1) Z1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS X (CONS (CONS C V1) Z1)))) (NOT (MEMBER F D)) V (NUMBERP W) X (CAR X) (NOT (NUMBERP (CDR X)))) (EQUAL C Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 2.1.3. (IMPLIES (AND (NOT (LISTP C)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) C)) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) C))) (NOT (MEMBER F D)) V (NUMBERP W) X1 (CAR C) (NOT (NUMBERP Z1))) (EQUAL (CAAR C) Z)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the function STRIP-CARS, to: T. Case 2.1.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS X V1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS X V1)))) (NOT (MEMBER F D)) V (NUMBERP W) X1 X (NOT (NUMBERP Z1))) (EQUAL (CAR X) Z)). This finally simplifies, rewriting with CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding the functions STRIP-CARS, MEMBER, and EQUAL, to: T. Case 2.1.1. (IMPLIES (AND (EQUAL (STRIP-CARS (CONS (CONS X1 Z1) (CONS (CONS C W1) V1))) (CONS V (CONS Z (STRIP-CARS D)))) (NOT (MEMBER F (CONS (CONS X1 Z1) (CONS (CONS C W1) V1)))) (NOT (MEMBER F D)) V (NUMBERP W) X1 (NOT (NUMBERP Z1))) (EQUAL C Z)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up STRIP-CARS, to: T. Case 1. (IMPLIES (AND (NOT (EQUAL FN 'QUOTE)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (EQUAL FN 'IF) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAR (CONS (CAADR ARGS1) (ADD1 (PLUS (CDAR ARGS1) (CDADR ARGS1))))) (CAR (CONS (CAADR ARGS2) (ADD1 (PLUS (CDAR ARGS2) (CDADR ARGS2))))))). However this again simplifies, applying CAR-CONS, and expanding EQUAL, to: (IMPLIES (AND (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADR ARGS2))). Applying the lemma CAR-CDR-ELIM, replace ARGS2 by (CONS X Z) to eliminate (CAR ARGS2) and (CDR ARGS2), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We thus obtain the following seven new formulas: Case 1.7. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS ARGS2)) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F ARGS2)) (CAR ARGS2) (CAAR ARGS2) (CADR ARGS2) (NUMBERP (CDAR ARGS2)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAADR ARGS2))). But this further simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and opening up STRIP-CARS, MEMBER, CAR, CDR, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X Z) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS V W) to eliminate (CAR X) and (CDR X), Z by (CONS X D) to eliminate (CAR Z) and (CDR Z), X by (CONS Z C) to eliminate (CAR X) and (CDR X), Z by (CONS V W) to eliminate (CAR Z) and (CDR Z), and V by (CONS Z D) to eliminate (CAR V) and (CDR V). We would thus like to prove the following seven new conjectures: Case 1.7.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS ARGS1) NIL) (NOT (MEMBER F ARGS1)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). This finally simplifies, rewriting with CAR-NLISTP and CDR-NLISTP, and expanding STRIP-CARS, EQUAL, MEMBER, CAR, CDR, and NUMBERP, to: T. Case 1.7.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X Z)) NIL) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X))) (EQUAL (CAAR Z) 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS, CONS, and EQUAL, to: T. Case 1.7.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS V W))) NIL) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X))) (EQUAL (CAR V) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the function STRIP-CARS, to: T. Case 1.7.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS Z D) W))) NIL) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NUMBERP (CDR X))) (EQUAL Z 0)). However this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 1.7.3. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) Z)) NIL) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W)) (EQUAL (CAAR Z) 0)), which finally simplifies, rewriting with CDR-CONS and CAR-CONS, and unfolding STRIP-CARS, to: T. Case 1.7.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS X D))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W)) (EQUAL (CAR X) 0)). However this finally simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and expanding the function STRIP-CARS, to: T. Case 1.7.1. (IMPLIES (AND (NOT (LISTP ARGS2)) (EQUAL (STRIP-CARS (CONS (CONS V W) (CONS (CONS Z C) D))) NIL) (NOT (MEMBER F (CONS (CONS V W) (CONS (CONS Z C) D)))) V (NUMBERP W)) (EQUAL Z 0)). This finally simplifies, applying CDR-CONS and CAR-CONS, and expanding STRIP-CARS, to: T. Case 1.6. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X Z))) X (CAR X) (CAR Z) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAAR Z))). This further simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding the definitions of STRIP-CARS, CONS, MEMBER, NUMBERP, and CAR, to: (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Appealing to the lemma CAR-CDR-ELIM, we now replace ARGS1 by (CONS V W) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS D C) to eliminate (CAR V) and (CDR V), W by (CONS V X1) to eliminate (CAR W) and (CDR W), V by (CONS W Z1) to eliminate (CAR V) and (CDR V), W by (CONS D C) to eliminate (CAR W) and (CDR W), and D by (CONS W X1) to eliminate (CAR D) and (CDR D). The result is seven new goals: Case 1.6.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) '(0)) (NOT (MEMBER F ARGS1)) X (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)), which finally simplifies, unfolding the definitions of STRIP-CARS and EQUAL, to: T. Case 1.6.6. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V W)) '(0)) (NOT (MEMBER F (CONS V W))) X V (CAR V) (CAR W) (NUMBERP (CDR V))) (EQUAL (CAAR W) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, CONS, EQUAL, MEMBER, NUMBERP, and CAR, to: T. Case 1.6.5. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS D C))) '(0)) (NOT (MEMBER F (CONS V (CONS D C)))) X V (CAR V) D (NUMBERP (CDR V))) (EQUAL (CAR D) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 1.6.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS W X1) C))) '(0)) (NOT (MEMBER F (CONS V (CONS (CONS W X1) C)))) X V (CAR V) (NUMBERP (CDR V))) (EQUAL W 0)). But this finally simplifies, applying the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS, CAR, EQUAL, and CDR, to: T. Case 1.6.3. (IMPLIES (AND (NOT (LISTP W)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) W)) '(0)) (NOT (MEMBER F (CONS (CONS D C) W))) X D (CAR W) (NUMBERP C)) (EQUAL (CAAR W) 0)), which finally simplifies, applying CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, CAR, EQUAL, and MEMBER, to: T. Case 1.6.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS V X1))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS V X1)))) X D V (NUMBERP C)) (EQUAL (CAR V) 0)). But this finally simplifies, rewriting with CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding the definitions of STRIP-CARS and CAR, to: T. Case 1.6.1. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS D C) (CONS (CONS W Z1) X1))) '(0)) (NOT (MEMBER F (CONS (CONS D C) (CONS (CONS W Z1) X1)))) X D (NUMBERP C)) (EQUAL W 0)). However this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the functions STRIP-CARS and CAR, to: T. Case 1.5. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS V W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS V W)))) X (CAR X) V (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAR V))). However this further simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and unfolding the definitions of STRIP-CARS, MEMBER, and NUMBERP, to: (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS Z D) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS C X1) to eliminate (CAR Z) and (CDR Z), D by (CONS Z Z1) to eliminate (CAR D) and (CDR D), Z by (CONS D V1) to eliminate (CAR Z) and (CDR Z), D by (CONS C X1) to eliminate (CAR D) and (CDR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). This produces the following seven new formulas: Case 1.5.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X V (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). But this finally simplifies, expanding the definition of STRIP-CARS, to: T. Case 1.5.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z D)) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z D))) (NOT (MEMBER F W)) X V Z (CAR Z) (CAR D) (NUMBERP (CDR Z))) (EQUAL (CAAR D) 0)), which finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 1.5.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS C X1))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS C X1)))) (NOT (MEMBER F W)) X V Z (CAR Z) C (NUMBERP (CDR Z))) (EQUAL (CAR C) 0)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding the functions STRIP-CARS, EQUAL, MEMBER, and NUMBERP, to: T. Case 1.5.4. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS Z (CONS (CONS D Z1) X1))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS Z (CONS (CONS D Z1) X1)))) (NOT (MEMBER F W)) X V Z (CAR Z) (NUMBERP (CDR Z))) (EQUAL D 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and opening up the functions STRIP-CARS and EQUAL, to: T. Case 1.5.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X V C (CAR D) (NUMBERP X1)) (EQUAL (CAAR D) 0)), which finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding the definition of STRIP-CARS, to: T. Case 1.5.2. (IMPLIES (AND (NOT (LISTP Z)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS Z Z1))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS Z Z1)))) (NOT (MEMBER F W)) X V C Z (NUMBERP X1)) (EQUAL (CAR Z) 0)), which finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and opening up STRIP-CARS, EQUAL, and MEMBER, to: T. Case 1.5.1. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS D V1) Z1))) (CONS 0 (CONS 0 (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS D V1) Z1)))) (NOT (MEMBER F W)) X V C (NUMBERP X1)) (EQUAL D 0)). But this finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and opening up the function STRIP-CARS, to: T. Case 1.4. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS X (CONS (CONS Z D) W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS X (CONS (CONS Z D) W)))) X (CAR X) (NUMBERP (CDR X)) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) Z)). But this further simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CDR-NLISTP, and expanding STRIP-CARS, MEMBER, and NUMBERP, to the new conjecture: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) Z)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS V D) to eliminate (CAR ARGS1) and (CDR ARGS1), V by (CONS C X1) to eliminate (CAR V) and (CDR V), D by (CONS V Z1) to eliminate (CAR D) and (CDR D), V by (CONS D V1) to eliminate (CAR V) and (CDR V), D by (CONS C X1) to eliminate (CAR D) and (CDR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following seven new formulas: Case 1.4.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F W)) X (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) Z)). However this finally simplifies, unfolding the function STRIP-CARS, to: T. Case 1.4.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V D))) (NOT (MEMBER F W)) X V (CAR V) (CAR D) (NUMBERP (CDR V))) (EQUAL (CAAR D) Z)), which finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, and CAR-CONS, and unfolding the functions STRIP-CARS, CONS, CAR, EQUAL, and CDR, to: T. Case 1.4.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS C X1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS C X1)))) (NOT (MEMBER F W)) X V (CAR V) C (NUMBERP (CDR V))) (EQUAL (CAR C) Z)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and expanding the definitions of STRIP-CARS, EQUAL, MEMBER, and NUMBERP, to: T. Case 1.4.4. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS V (CONS (CONS D Z1) X1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS V (CONS (CONS D Z1) X1)))) (NOT (MEMBER F W)) X V (CAR V) (NUMBERP (CDR V))) (EQUAL D Z)). However this finally simplifies, applying CDR-CONS, CAR-NLISTP, and CAR-CONS, and expanding the definitions of STRIP-CARS and EQUAL, to: T. Case 1.4.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) D))) (NOT (MEMBER F W)) X C (CAR D) (NUMBERP X1)) (EQUAL (CAAR D) Z)). But this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, to: T. Case 1.4.2. (IMPLIES (AND (NOT (LISTP V)) (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS V Z1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS V Z1)))) (NOT (MEMBER F W)) X C V (NUMBERP X1)) (EQUAL (CAR V) Z)), which finally simplifies, applying the lemmas CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and opening up the definitions of STRIP-CARS, MEMBER, and EQUAL, to: T. Case 1.4.1. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS D V1) Z1))) (CONS 0 (CONS Z (STRIP-CARS W)))) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS D V1) Z1)))) (NOT (MEMBER F W)) X C (NUMBERP X1)) (EQUAL D Z)), which finally simplifies, applying CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding the function STRIP-CARS, to: T. Case 1.3. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) Z))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) Z))) V (CAR Z) (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAAR Z))). This further simplifies, applying CDR-CONS, CAR-CONS, and CAR-NLISTP, and opening up the definitions of STRIP-CARS, MEMBER, and CAR, to the new conjecture: (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS X D) to eliminate (CAR ARGS1) and (CDR ARGS1), X by (CONS C X1) to eliminate (CAR X) and (CDR X), D by (CONS X Z1) to eliminate (CAR D) and (CDR D), X by (CONS D V1) to eliminate (CAR X) and (CDR X), D by (CONS C X1) to eliminate (CAR D) and (CDR D), and C by (CONS D Z1) to eliminate (CAR C) and (CDR C). We would thus like to prove the following seven new conjectures: Case 1.3.7. (IMPLIES (AND (NOT (LISTP ARGS1)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS ARGS1) (LIST V)) (NOT (MEMBER F ARGS1)) V (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). But this finally simplifies, opening up the definition of STRIP-CARS, to: T. Case 1.3.6. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X D)) (LIST V)) (NOT (MEMBER F (CONS X D))) V (NUMBERP W) X (CAR X) (CAR D) (NUMBERP (CDR X))) (EQUAL (CAAR D) 0)), which finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, CONS-EQUAL, and CDR-NLISTP, and unfolding STRIP-CARS, CONS, CAR, EQUAL, MEMBER, and NUMBERP, to: T. Case 1.3.5. (IMPLIES (AND (NOT (LISTP C)) (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS C X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS C X1)))) V (NUMBERP W) X (CAR X) C (NUMBERP (CDR X))) (EQUAL (CAR C) 0)). This finally simplifies, rewriting with CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, to: T. Case 1.3.4. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS X (CONS (CONS D Z1) X1))) (LIST V)) (NOT (MEMBER F (CONS X (CONS (CONS D Z1) X1)))) V (NUMBERP W) X (CAR X) (NUMBERP (CDR X))) (EQUAL D 0)). However this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-NLISTP, CAR-CONS, and CONS-EQUAL, and expanding STRIP-CARS, to: T. Case 1.3.3. (IMPLIES (AND (NOT (LISTP D)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) D)) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) D))) V (NUMBERP W) C (CAR D) (NUMBERP X1)) (EQUAL (CAAR D) 0)), which finally simplifies, applying CDR-CONS, CAR-CONS, CONS-EQUAL, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS, EQUAL, MEMBER, and CAR, to: T. Case 1.3.2. (IMPLIES (AND (NOT (LISTP X)) (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS X Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS X Z1)))) V (NUMBERP W) C X (NUMBERP X1)) (EQUAL (CAR X) 0)). This finally simplifies, applying CDR-CONS, CAR-CONS, CAR-NLISTP, and CONS-EQUAL, and unfolding the definition of STRIP-CARS, to: T. Case 1.3.1. (IMPLIES (AND (NOT (LISTP Z)) (EQUAL (STRIP-CARS (CONS (CONS C X1) (CONS (CONS D V1) Z1))) (LIST V)) (NOT (MEMBER F (CONS (CONS C X1) (CONS (CONS D V1) Z1)))) V (NUMBERP W) C (NUMBERP X1)) (EQUAL D 0)). But this finally simplifies, rewriting with the lemmas CDR-CONS, CAR-CONS, and CONS-EQUAL, and unfolding STRIP-CARS, to: T. Case 1.2. (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (STRIP-CARS (CONS (CONS V W) (CONS X D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F (CONS (CONS V W) (CONS X D)))) V X (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) (CAR X))), which further simplifies, rewriting with CDR-CONS, CAR-CONS, and CAR-NLISTP, and unfolding the definitions of STRIP-CARS and MEMBER, to the new conjecture: (IMPLIES (AND (NOT (LISTP X)) (EQUAL (STRIP-CARS ARGS1) (CONS V (CONS 0 (STRIP-CARS D)))) (NOT (MEMBER F ARGS1)) (NOT (MEMBER F D)) V X (NUMBERP W) (CAR ARGS1) (CAAR ARGS1) (CADR ARGS1) (NUMBERP (CDAR ARGS1))) (EQUAL (CAADR ARGS1) 0)). Applying the lemma CAR-CDR-ELIM, replace ARGS1 by (CONS Z C) to eliminate (CAR ARGS1) and (CDR ARGS1), Z by (CONS X1 Z1) to eliminate (CAR Z) and (CDR Z), C by