(BOOT-STRAP NQTHM) [ 0.1 0.1 0.0 ] GROUND-ZERO (COMPILE-UNCOMPILED-DEFNS "tmp") Loading ./basic/tmp.o Finished loading ./basic/tmp.o /v/hank/v28/boyer/nqthm-2nd/nqthm-1992/examples/basic/tmp.lisp (ADD-SHELL ZN NIL ZNP ((POS (ONE-OF NUMBERP) ZERO) (NEG (ONE-OF NUMBERP) ZERO))) [ 0.1 0.0 0.0 ] ZN (DEFN ZLESSP (X Y) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) From the definition we can conclude that: (OR (FALSEP (ZLESSP X Y)) (TRUEP (ZLESSP X Y))) is a theorem. [ 0.0 0.0 0.0 ] ZLESSP (DEFN ZLESSEQP (X Y) (NOT (ZLESSP Y X))) Note that (OR (FALSEP (ZLESSEQP X Y)) (TRUEP (ZLESSEQP X Y))) is a theorem. [ 0.0 0.0 0.0 ] ZLESSEQP (DEFN ZMAX (X Y) (IF (ZLESSP X Y) Y X)) Note that (OR (EQUAL (ZMAX X Y) X) (EQUAL (ZMAX X Y) Y)) is a theorem. [ 0.0 0.0 0.0 ] ZMAX (DEFN ZMIN (X Y) (IF (ZLESSP X Y) X Y)) Note that (OR (EQUAL (ZMIN X Y) X) (EQUAL (ZMIN X Y) Y)) is a theorem. [ 0.0 0.0 0.0 ] ZMIN (DEFN ZSUB1 (X) (ZN (POS X) (ADD1 (NEG X)))) From the definition we can conclude that (ZNP (ZSUB1 X)) is a theorem. [ 0.0 0.0 0.0 ] ZSUB1 (DEFN PZDIFFERENCE (X Y) (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) From the definition we can conclude that (NUMBERP (PZDIFFERENCE X Y)) is a theorem. [ 0.0 0.0 0.0 ] PZDIFFERENCE (DEFN M1 (X Y Z) (IF (ZLESSEQP X Y) 0 1)) WARNING: Z is in the arglist but not in the body of the definition of M1. Observe that (NUMBERP (M1 X Y Z)) is a theorem. [ 0.0 0.0 0.0 ] M1 (DEFN M2 (X Y Z) (PZDIFFERENCE (ZMAX X (ZMAX Y Z)) (ZMIN X (ZMIN Y Z)))) Note that (NUMBERP (M2 X Y Z)) is a theorem. [ 0.0 0.0 0.0 ] M2 (DEFN M3 (X Y Z) (PZDIFFERENCE X (ZMIN X (ZMIN Y Z)))) Note that (NUMBERP (M3 X Y Z)) is a theorem. [ 0.0 0.0 0.0 ] M3 (DEFN TAK0 (X Y Z) (IF (ZLESSEQP X Y) Y (IF (ZLESSEQP Y Z) Z X))) Observe that: (OR (EQUAL (TAK0 X Y Z) X) (EQUAL (TAK0 X Y Z) Y) (EQUAL (TAK0 X Y Z) Z)) is a theorem. [ 0.0 0.0 0.0 ] TAK0 (DEFN M (X Y Z) (CONS (M1 X Y Z) (CONS (M2 X Y Z) (CONS (M3 X Y Z) NIL)))) Note that (LISTP (M X Y Z)) is a theorem. [ 0.0 0.0 0.0 ] M (PROVE-LEMMA TAK0-SATISFIES-TAK-EQUATION NIL (EQUAL (TAK0 X Y Z) (IF (ZLESSEQP X Y) Y (TAK0 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y))))) This conjecture simplifies, rewriting with the lemmas POS-ZN and NEG-ZN, and unfolding the functions ZLESSEQP, ZLESSP, TAK0, and ZSUB1, to 24 new conjectures: Case 24.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Z X (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 23.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 (ZN (POS X) (ADD1 (NEG X))) X (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 22.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 Y X (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 21.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Y X (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 20.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Z (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 19.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 (ZN (POS X) (ADD1 (NEG X))) (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 18.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 Y (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 17.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Y (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 16.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Z Z (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, rewriting with the lemmas SUB1-ADD1, NEG-ZN, and POS-ZN, and expanding the functions LESSP, PLUS, ZLESSEQP, ZLESSP, and TAK0, to the conjecture: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Z))) (PLUS (NEG Z) (POS Z))))) (EQUAL Z (ZN (POS Z) (ADD1 (NEG Z))))). This again simplifies, using linear arithmetic, to: T. Case 15.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 (ZN (POS X) (ADD1 (NEG X))) Z (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 14.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 Y Z (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, using linear arithmetic, to: T. Case 13.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Y Z (ZN (POS Z) (ADD1 (NEG Z)))))), which again simplifies, unfolding the definitions of ZLESSEQP, ZLESSP, and TAK0, to: T. Case 12.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Z X X))), which again simplifies, using linear arithmetic, to: T. Case 11.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 (ZN (POS X) (ADD1 (NEG X))) X X))), which again simplifies, applying POS-ZN and NEG-ZN, and opening up the functions ZLESSEQP, ZLESSP, and TAK0, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X))) (LESSP (PLUS (POS X) (NEG X)) (PLUS (NEG X) (POS X)))) (EQUAL X (ZN (POS X) (ADD1 (NEG X))))), which again simplifies, using linear arithmetic, to: T. Case 10.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 Y X X))), which again simplifies, unfolding the functions ZLESSEQP, ZLESSP, and TAK0, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG X)) (PLUS (NEG X) (POS X)))) (EQUAL X Y)). However this again simplifies, using linear arithmetic, to: T. Case 9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Y X X))), which again simplifies, using linear arithmetic, to: T. Case 8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Z (ZN (POS Y) (ADD1 (NEG Y))) X))), which again simplifies, using linear arithmetic, to: T. Case 7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 (ZN (POS X) (ADD1 (NEG X))) (ZN (POS Y) (ADD1 (NEG Y))) X))), which again simplifies, using linear arithmetic, to: T. Case 6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 Y (ZN (POS Y) (ADD1 (NEG Y))) X))), which again simplifies, using linear arithmetic, to: T. Case 5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Y (ZN (POS Y) (ADD1 (NEG Y))) X))), which again simplifies, using linear arithmetic, to: T. Case 4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Z Z X))), which again simplifies, expanding the functions ZLESSEQP, ZLESSP, and TAK0, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (EQUAL Z X)). However this again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 (ZN (POS X) (ADD1 (NEG X))) Z X))), which again simplifies, applying POS-ZN and NEG-ZN, and unfolding ZLESSEQP, ZLESSP, and TAK0, to the following two new formulas: Case 3.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))))) (EQUAL X Z)). However this again simplifies, using linear arithmetic, to: T. Case 3.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (EQUAL X (ZN (POS X) (ADD1 (NEG X))))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (EQUAL X (TAK0 Y Z X))), which again simplifies, opening up the functions ZLESSEQP, ZLESSP, and TAK0, to the formula: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (EQUAL X Y)). But this again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))) (EQUAL Z (TAK0 Y Z X))), which again simplifies, unfolding ZLESSEQP, ZLESSP, and TAK0, to: T. Q.E.D. [ 0.0 0.0 0.1 ] TAK0-SATISFIES-TAK-EQUATION (PROVE-LEMMA M1-LESSEQP-0 (REWRITE) (IMPLIES (NOT (ZLESSEQP X Y)) (NOT (LESSP (M1 X Y Z) (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)))))) WARNING: Note that the linear lemma M1-LESSEQP-0 is being stored under the term (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)), which is unusual because M1 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M1-LESSEQP-0 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, and IMPLIES, to: (IMPLIES (ZLESSP Y X) (NOT (LESSP (M1 X Y Z) (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y))))), which simplifies, applying POS-ZN and NEG-ZN, and unfolding the functions ZLESSP, ZLESSEQP, M1, ZSUB1, and TAK0, to the following 18 new formulas: Case 18.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (ZLESSP X Z))) (NOT (LESSP 1 0))). This again simplifies, using linear arithmetic, to: T. Case 17.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Z))) (NOT (LESSP 1 0))), which again simplifies, using linear arithmetic, to: T. Case 16.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (ZLESSP Z Z))) (NOT (LESSP 1 0))), which again simplifies, using linear arithmetic, to: T. Case 15.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (ZLESSP X (ZN (POS X) (ADD1 (NEG X)))))) (NOT (LESSP 1 0))), which again simplifies, using linear arithmetic, to: T. Case 14.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS X) (ADD1 (NEG X)))))) (NOT (LESSP 1 0))), which again simplifies, using linear arithmetic, to: T. Case 13.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (ZLESSP Z (ZN (POS X) (ADD1 (NEG X)))))) (NOT (LESSP 1 0))), which again simplifies, using linear arithmetic, to: T. Case 12.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (ZLESSP X Y))) (NOT (LESSP 1 0))), which again simplifies, using linear arithmetic, to: T. Case 11.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Y))) (NOT (LESSP 1 0))), which again simplifies, using linear arithmetic, to: T. Case 10.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (ZLESSP Z Y))) (NOT (LESSP 1 0))), which again simplifies, using linear arithmetic, to: T. Case 9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Z)) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Case 8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Z)) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Case 7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Z)) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Case 6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X (ZN (POS X) (ADD1 (NEG X))))) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Case 5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS X) (ADD1 (NEG X))))) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Case 4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z (ZN (POS X) (ADD1 (NEG X))))) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Y)) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Y)) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Y)) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] M1-LESSEQP-0 (PROVE-LEMMA M1-LESSEQP-1 (REWRITE) (IMPLIES (NOT (ZLESSEQP X Y)) (NOT (LESSP (M1 X Y Z) (M1 (ZSUB1 X) Y Z))))) WARNING: Note that the linear lemma M1-LESSEQP-1 is being stored under the term (M1 (ZSUB1 X) Y Z), which is unusual because M1 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M1-LESSEQP-1 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, and IMPLIES, to the new conjecture: (IMPLIES (ZLESSP Y X) (NOT (LESSP (M1 X Y Z) (M1 (ZSUB1 X) Y Z)))), which simplifies, applying POS-ZN and NEG-ZN, and unfolding the functions ZLESSP, ZLESSEQP, M1, and ZSUB1, to the following two new goals: Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))))) (NOT (LESSP 1 0))). However this again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] M1-LESSEQP-1 (PROVE-LEMMA M1-LESSEQP-2 (REWRITE) (IMPLIES (NOT (ZLESSEQP X Y)) (NOT (LESSP (M1 X Y Z) (M1 (ZSUB1 Y) Z X))))) WARNING: Note that the linear lemma M1-LESSEQP-2 is being stored under the term (M1 (ZSUB1 Y) Z X), which is unusual because M1 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M1-LESSEQP-2 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, and IMPLIES, to the new conjecture: (IMPLIES (ZLESSP Y X) (NOT (LESSP (M1 X Y Z) (M1 (ZSUB1 Y) Z X)))), which simplifies, applying POS-ZN and NEG-ZN, and unfolding the functions ZLESSP, ZLESSEQP, M1, and ZSUB1, to the following two new goals: Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))))) (NOT (LESSP 1 0))). However this again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] M1-LESSEQP-2 (PROVE-LEMMA M1-LESSEQP-3 (REWRITE) (IMPLIES (NOT (ZLESSEQP X Y)) (NOT (LESSP (M1 X Y Z) (M1 (ZSUB1 Z) X Y))))) WARNING: Note that the linear lemma M1-LESSEQP-3 is being stored under the term (M1 (ZSUB1 Z) X Y), which is unusual because M1 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M1-LESSEQP-3 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, and IMPLIES, to the new conjecture: (IMPLIES (ZLESSP Y X) (NOT (LESSP (M1 X Y Z) (M1 (ZSUB1 Z) X Y)))), which simplifies, applying POS-ZN and NEG-ZN, and unfolding the functions ZLESSP, ZLESSEQP, M1, and ZSUB1, to the following two new goals: Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (NOT (LESSP 1 0))). However this again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP 1 1))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.0 ] M1-LESSEQP-3 (PROVE-LEMMA M2-LESSEQP-0 (REWRITE) (IMPLIES (NOT (ZLESSEQP X Y)) (NOT (LESSP (M2 X Y Z) (M2 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)))))) WARNING: Note that the linear lemma M2-LESSEQP-0 is being stored under the term (M2 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)), which is unusual because M2 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M2-LESSEQP-0 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, and IMPLIES, to: (IMPLIES (ZLESSP Y X) (NOT (LESSP (M2 X Y Z) (M2 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y))))), which simplifies, applying POS-ZN and NEG-ZN, and unfolding the functions ZLESSP, ZMIN, ZMAX, M2, ZSUB1, ZLESSEQP, and TAK0, to the following 36 new formulas: Case 36.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Y Z X)))). This again simplifies, opening up ZLESSP, ZMAX, ZMIN, and M2, to eight new goals: Case 36.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y X))))), which again simplifies, using linear arithmetic, to: T. Case 36.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y X))))), which again simplifies, using linear arithmetic, to: T. Case 36.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y X))))), which again simplifies, using linear arithmetic, to: T. Case 36.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y X))))), which again simplifies, using linear arithmetic, to: T. Case 36.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZMAX Y X) (ZMIN Y Z))))), which again simplifies, opening up the definitions of PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to the formula: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))))). But this again simplifies, using linear arithmetic, to: T. Case 36.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZMAX Y X) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 36.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Y X) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 36.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZMAX Y X) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 35.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Y Z X)))), which again simplifies, expanding the definitions of ZLESSP, ZMAX, ZMIN, and M2, to eight new formulas: Case 35.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y X))))), which again simplifies, expanding PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). However this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). This again simplifies, using linear arithmetic, to: T. Case 35.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y X))))), which again simplifies, unfolding the definitions of PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to the goal: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). However this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). However this again simplifies, using linear arithmetic, to: T. Case 35.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y X))))), which again simplifies, using linear arithmetic, to: T. Case 35.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y X))))), which again simplifies, using linear arithmetic, to: T. Case 35.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZMAX Y X) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 35.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZMAX Y X) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 35.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Y X) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 35.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZMAX Y X) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 34.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 (ZN (POS X) (ADD1 (NEG X))) Z X)))), which again simplifies, opening up the functions ZLESSP, ZMAX, ZMIN, and M2, to eight new conjectures: Case 34.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) Z) (ZMIN (ZN (POS X) (ADD1 (NEG X))) X))))), which again simplifies, using linear arithmetic, to: T. Case 34.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) Z) (ZMIN (ZN (POS X) (ADD1 (NEG X))) X))))), which again simplifies, using linear arithmetic, to: T. Case 34.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) Z) (ZMIN (ZN (POS X) (ADD1 (NEG X))) X))))), which again simplifies, using linear arithmetic, to: T. Case 34.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) Z) (ZMIN (ZN (POS X) (ADD1 (NEG X))) X))))), which again simplifies, using linear arithmetic, to: T. Case 34.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) X) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Z))))), which again simplifies, applying POS-ZN, NEG-ZN, and SUB1-ADD1, and expanding the definitions of PZDIFFERENCE, ZLESSP, PLUS, LESSP, ZMAX, and ZMIN, to the following nine new goals: Case 34.4.9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE X (ZN (POS X) (ADD1 (NEG X))))))). But this again simplifies, using linear arithmetic, to: T. Case 34.4.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X)))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE X (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 34.4.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE X (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 34.4.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X)))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE X (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 34.4.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 34.4.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 34.4.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X)))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE X Z)))), which again simplifies, expanding the function PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X)))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))))). But this again simplifies, using linear arithmetic, to: T. Case 34.4.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE X Z)))), which again simplifies, unfolding the function PZDIFFERENCE, to the goal: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))))). This again simplifies, using linear arithmetic, to: T. Case 34.4.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Z)))), which again simplifies, using linear arithmetic, to: T. Case 34.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) X) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Z))))), which again simplifies, using linear arithmetic, to: T. Case 34.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) X) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Z))))), which again simplifies, using linear arithmetic, to: T. Case 34.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) X) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Z))))), which again simplifies, using linear arithmetic, to: T. Case 33.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 (ZN (POS X) (ADD1 (NEG X))) Z X)))), which again simplifies, using linear arithmetic, to: T. Case 32.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Z Z X)))), which again simplifies, opening up the definitions of ZLESSP, ZMAX, ZMIN, and M2, to eight new goals: Case 32.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z X))))), which again simplifies, using linear arithmetic, to: T. Case 32.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z X))))), which again simplifies, using linear arithmetic, to: T. Case 32.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z X))))), which again simplifies, using linear arithmetic, to: T. Case 32.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z X))))), which again simplifies, using linear arithmetic, to: T. Case 32.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZMAX Z X) (ZMIN Z Z))))), which again simplifies, unfolding PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to the goal: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))))). This again simplifies, using linear arithmetic, to: T. Case 32.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZMAX Z X) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 32.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Z X) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 32.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZMAX Z X) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 31.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Z Z X)))), which again simplifies, opening up the definitions of ZLESSP, ZMAX, ZMIN, and M2, to eight new formulas: Case 31.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z X))))), which again simplifies, unfolding PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))))). However this again simplifies, using linear arithmetic, to: T. Case 31.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z X))))), which again simplifies, expanding the functions PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))))). But this again simplifies, using linear arithmetic, to: T. Case 31.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z X))))), which again simplifies, using linear arithmetic, to: T. Case 31.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z X))))), which again simplifies, using linear arithmetic, to: T. Case 31.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZMAX Z X) (ZMIN Z Z))))), which again simplifies, expanding PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))))). But this again simplifies, using linear arithmetic, to: T. Case 31.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZMAX Z X) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 31.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Z X) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 31.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZMAX Z X) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 30.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Y (ZN (POS Y) (ADD1 (NEG Y))) X)))), which again simplifies, using linear arithmetic, to: T. Case 29.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Y (ZN (POS Y) (ADD1 (NEG Y))) X)))), which again simplifies, using linear arithmetic, to: T. Case 28.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 (ZN (POS X) (ADD1 (NEG X))) (ZN (POS Y) (ADD1 (NEG Y))) X)))), which again simplifies, using linear arithmetic, to: T. Case 27.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 (ZN (POS X) (ADD1 (NEG X))) (ZN (POS Y) (ADD1 (NEG Y))) X)))), which again simplifies, using linear arithmetic, to: T. Case 26.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Z (ZN (POS Y) (ADD1 (NEG Y))) X)))), which again simplifies, using linear arithmetic, to: T. Case 25.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Z (ZN (POS Y) (ADD1 (NEG Y))) X)))), which again simplifies, using linear arithmetic, to: T. Case 24.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Y X X)))), which again simplifies, unfolding the functions ZLESSP, ZMAX, ZMIN, PZDIFFERENCE, and M2, to two new goals: Case 24.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))))), which again simplifies, expanding the definition of PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))))). This again simplifies, using linear arithmetic, to: T. Case 24.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))))), which again simplifies, using linear arithmetic, to: T. Case 23.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Y X X)))), which again simplifies, using linear arithmetic, to: T. Case 22.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 (ZN (POS X) (ADD1 (NEG X))) X X)))), which again simplifies, appealing to the lemmas SUB1-ADD1, NEG-ZN, and POS-ZN, and opening up the definitions of ZLESSP, ZMAX, ZMIN, LESSP, PLUS, and M2, to six new conjectures: Case 22.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE X (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, appealing to the lemmas POS-ZN and NEG-ZN, and unfolding the function PZDIFFERENCE, to the conjecture: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X)))))). However this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X))) (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X)))))). However this again simplifies, using linear arithmetic, to: T. Case 22.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE X (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, rewriting with POS-ZN and NEG-ZN, and unfolding the definition of PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X)))))), which again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X))) (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X)))))). This again simplifies, using linear arithmetic, to: T. Case 22.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE X (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 22.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE X (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 22.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) X)))), which again simplifies, using linear arithmetic, to: T. Case 22.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) X)))), which again simplifies, using linear arithmetic, to: T. Case 21.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 (ZN (POS X) (ADD1 (NEG X))) X X)))), which again simplifies, using linear arithmetic, to: T. Case 20.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Z X X)))), which again simplifies, using linear arithmetic, to: T. Case 19.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Z X X)))), which again simplifies, using linear arithmetic, to: T. Case 18.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Y Z (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 17.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Y Z (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, appealing to the lemmas POS-ZN and NEG-ZN, and unfolding the definitions of ZLESSP, ZMAX, ZMIN, and M2, to eight new conjectures: Case 17.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y (ZN (POS Z) (ADD1 (NEG Z)))))))), which again simplifies, using linear arithmetic, to: T. Case 17.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y (ZN (POS Z) (ADD1 (NEG Z)))))))), which again simplifies, applying the lemmas NEG-ZN and POS-ZN, and opening up the definitions of PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to two new formulas: Case 17.7.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG Z))) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (PZDIFFERENCE Z (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 17.7.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG Z))) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (PZDIFFERENCE Z Y)))), which again simplifies, unfolding PZDIFFERENCE, to the goal: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG Z))) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). But this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG Z))) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). This again simplifies, using linear arithmetic, to: T. Case 17.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y (ZN (POS Z) (ADD1 (NEG Z)))))))), which again simplifies, using linear arithmetic, to: T. Case 17.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZMAX Y Z) (ZMIN Y (ZN (POS Z) (ADD1 (NEG Z)))))))), which again simplifies, using linear arithmetic, to: T. Case 17.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZMAX Y (ZN (POS Z) (ADD1 (NEG Z)))) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 17.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZMAX Y (ZN (POS Z) (ADD1 (NEG Z)))) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 17.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Y (ZN (POS Z) (ADD1 (NEG Z)))) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 17.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZMAX Y (ZN (POS Z) (ADD1 (NEG Z)))) (ZMIN Y Z))))), which again simplifies, using linear arithmetic, to: T. Case 16.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 (ZN (POS X) (ADD1 (NEG X))) Z (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 15.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 (ZN (POS X) (ADD1 (NEG X))) Z (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 14.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Z Z (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 13.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Z Z (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, rewriting with POS-ZN and NEG-ZN, and unfolding ZLESSP, ZMAX, ZMIN, and M2, to the following eight new goals: Case 13.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))))). However this again simplifies, using linear arithmetic, to: T. Case 13.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))))), which again simplifies, applying NEG-ZN and POS-ZN, and opening up the definitions of PZDIFFERENCE, ZLESSP, ZMAX, and ZMIN, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (DIFFERENCE (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))))), which again simplifies, using linear arithmetic, to: T. Case 13.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))))), which again simplifies, using linear arithmetic, to: T. Case 13.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZMAX Z Z) (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))))), which again simplifies, using linear arithmetic, to: T. Case 13.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZMAX Z (ZN (POS Z) (ADD1 (NEG Z)))) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 13.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZMAX Z (ZN (POS Z) (ADD1 (NEG Z)))) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 13.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZMAX Z (ZN (POS Z) (ADD1 (NEG Z)))) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 13.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZMAX Z (ZN (POS Z) (ADD1 (NEG Z)))) (ZMIN Z Z))))), which again simplifies, using linear arithmetic, to: T. Case 12.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Y (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 11.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Y (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 10.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 (ZN (POS X) (ADD1 (NEG X))) (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 (ZN (POS X) (ADD1 (NEG X))) (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Z (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Z (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Y X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Y X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 (ZN (POS X) (ADD1 (NEG X))) X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 (ZN (POS X) (ADD1 (NEG X))) X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (M2 Z X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (M2 Z X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.5 0.1 ] M2-LESSEQP-0 (PROVE-LEMMA M2-LESSEQP-1 (REWRITE) (IMPLIES (NOT (ZLESSEQP X Y)) (NOT (LESSP (M2 X Y Z) (M2 (ZSUB1 X) Y Z))))) WARNING: Note that the linear lemma M2-LESSEQP-1 is being stored under the term (M2 (ZSUB1 X) Y Z), which is unusual because M2 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M2-LESSEQP-1 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, and IMPLIES, to the new conjecture: (IMPLIES (ZLESSP Y X) (NOT (LESSP (M2 X Y Z) (M2 (ZSUB1 X) Y Z)))), which simplifies, expanding ZLESSP, ZMIN, ZMAX, M2, and ZSUB1, to two new goals: Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) Y) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Z))))), which again simplifies, rewriting with POS-ZN, NEG-ZN, and SUB1-ADD1, and opening up the functions ZLESSP, ZMAX, ZMIN, PLUS, and LESSP, to the following 36 new conjectures: Case 2.36. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))). But this again simplifies, using linear arithmetic, to: T. Case 2.35. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.34. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.33. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.32. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.31. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.30. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.29. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.28. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.27. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.26. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.25. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.24. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.23. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.22. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.21. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE Y (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.20. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.19. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.18. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.17. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.16. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.15. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.14. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.13. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 2.12. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE Y Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.11. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE Y Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.10. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE Y Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE Y Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Y Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Y Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE Y Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE Y Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Z)))), which again simplifies, applying SUB1-ADD1, NEG-ZN, and POS-ZN, and unfolding the functions PZDIFFERENCE, DIFFERENCE, and PLUS, to the new formula: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (DIFFERENCE (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))))), which again simplifies, using linear arithmetic, to: T. Case 2.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Z)))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (PZDIFFERENCE (ZMAX (ZN (POS X) (ADD1 (NEG X))) Z) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Y))))), which again simplifies, applying the lemmas POS-ZN, NEG-ZN, and SUB1-ADD1, and expanding ZLESSP, ZMAX, ZMIN, PLUS, and LESSP, to 36 new goals: Case 1.36. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.35. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.34. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.33. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.32. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.31. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.30. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.29. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.28. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.27. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.26. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.25. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.24. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.23. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.22. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.21. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE Z (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.20. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.19. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.18. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.17. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.16. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.15. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.14. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.13. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS X) (NEG Y)) 0) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X))))))), which again simplifies, using linear arithmetic, to: T. Case 1.12. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE Z Y)))), which again simplifies, expanding the definition of PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). However this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). But this again simplifies, using linear arithmetic, to: T. Case 1.11. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE Z Y)))), which again simplifies, opening up the definitions of EQUAL, LESSP, and PZDIFFERENCE, to the conjecture: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (NEG X) (POS Z)) 0)) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). However this again simplifies, using linear arithmetic, to: (IMPLIES (AND (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (NEG X) (POS Z)) 0)) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))))). This again simplifies, using linear arithmetic, to: T. Case 1.10. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE Z Y)))), which again simplifies, using linear arithmetic, to: T. Case 1.9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE Z Y)))), which again simplifies, using linear arithmetic, to: T. Case 1.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Z Y)))), which again simplifies, using linear arithmetic, to: T. Case 1.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE Z Y)))), which again simplifies, using linear arithmetic, to: T. Case 1.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE Z Y)))), which again simplifies, using linear arithmetic, to: T. Case 1.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Z)) 0) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE Z Y)))), which again simplifies, using linear arithmetic, to: T. Case 1.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Y)))), which again simplifies, rewriting with SUB1-ADD1, NEG-ZN, and POS-ZN, and expanding PZDIFFERENCE, DIFFERENCE, and PLUS, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (DIFFERENCE (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))))), which again simplifies, using linear arithmetic, to: T. Case 1.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Y)))), which again simplifies, using linear arithmetic, to: T. Case 1.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Y)))), which again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z X) (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Y)))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.1 0.1 ] M2-LESSEQP-1 (PROVE-LEMMA M2-LESSEQP-2 (REWRITE) (IMPLIES (AND (NOT (ZLESSEQP X Y)) (EQUAL (M1 (ZSUB1 Y) Z X) (M1 X Y Z))) (NOT (LESSP (M2 X Y Z) (M2 (ZSUB1 Y) Z X))))) WARNING: Note that the linear lemma M2-LESSEQP-2 is being stored under the term (M2 (ZSUB1 Y) Z X), which is unusual because M2 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M2-LESSEQP-2 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, AND, and IMPLIES, to: (IMPLIES (AND (ZLESSP Y X) (EQUAL (M1 (ZSUB1 Y) Z X) (M1 X Y Z))) (NOT (LESSP (M2 X Y Z) (M2 (ZSUB1 Y) Z X)))), which simplifies, applying POS-ZN and NEG-ZN, and unfolding the functions ZLESSP, ZSUB1, ZLESSEQP, M1, ZMIN, ZMAX, and M2, to the following four new formulas: Case 4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (PZDIFFERENCE (ZMAX (ZN (POS Y) (ADD1 (NEG Y))) Z) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X))))). This again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (PZDIFFERENCE (ZMAX (ZN (POS Y) (ADD1 (NEG Y))) Z) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X))))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (PZDIFFERENCE (ZMAX (ZN (POS Y) (ADD1 (NEG Y))) X) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) Z))))), which again simplifies, applying the lemmas POS-ZN, NEG-ZN, and SUB1-ADD1, and opening up the definitions of ZLESSP, ZMAX, ZMIN, PLUS, and LESSP, to 36 new conjectures: Case 2.36. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (EQUAL (PLUS (POS Y) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.35. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.34. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Y) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.33. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.32. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (EQUAL (PLUS (POS Y) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.31. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.30. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Y) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.29. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.28. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (EQUAL (PLUS (POS Y) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.27. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.26. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Y) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.25. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.24. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (EQUAL (PLUS (POS Y) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.23. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.22. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Y) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.21. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE X (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.20. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.19. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.18. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.17. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.16. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.15. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.14. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.13. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y))))))), which again simplifies, using linear arithmetic, to: T. Case 2.12. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.11. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.10. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X Z) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y Z) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (NOT (LESSP (PZDIFFERENCE X X) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) Z)))), which again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PZDIFFERENCE Y X) (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) Z)))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (PZDIFFERENCE (ZMAX (ZN (POS Y) (ADD1 (NEG Y))) X) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) Z))))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.1 0.1 ] M2-LESSEQP-2 (PROVE-LEMMA M2-LESSEQP-3 (REWRITE) (IMPLIES (AND (NOT (ZLESSEQP X Y)) (EQUAL (M1 (ZSUB1 Z) X Y) (M1 X Y Z))) (NOT (LESSP (M2 X Y Z) (M2 (ZSUB1 Z) X Y))))) WARNING: Note that the linear lemma M2-LESSEQP-3 is being stored under the term (M2 (ZSUB1 Z) X Y), which is unusual because M2 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M2-LESSEQP-3 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, AND, and IMPLIES, to: (IMPLIES (AND (ZLESSP Y X) (EQUAL (M1 (ZSUB1 Z) X Y) (M1 X Y Z))) (NOT (LESSP (M2 X Y Z) (M2 (ZSUB1 Z) X Y)))), which simplifies, applying POS-ZN and NEG-ZN, and unfolding the functions ZLESSP, ZSUB1, ZLESSEQP, M1, ZMIN, ZMAX, and M2, to the following four new formulas: Case 4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (PZDIFFERENCE (ZMAX (ZN (POS Z) (ADD1 (NEG Z))) X) (ZMIN (ZN (POS Z) (ADD1 (NEG Z))) Y))))). This again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (PZDIFFERENCE (ZMAX (ZN (POS Z) (ADD1 (NEG Z))) X) (ZMIN (ZN (POS Z) (ADD1 (NEG Z))) Y))))), which again simplifies, applying POS-ZN, NEG-ZN, and SUB1-ADD1, and opening up the functions ZLESSP, ZMAX, ZMIN, PLUS, and LESSP, to the following 18 new goals: Case 3.18. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (EQUAL (PLUS (POS Z) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z))))))). But this again simplifies, using linear arithmetic, to: T. Case 3.17. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.16. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (EQUAL (PLUS (POS Z) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.15. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.14. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (EQUAL (PLUS (POS Z) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.13. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.12. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (EQUAL (PLUS (POS Z) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.11. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.10. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) (ZN (POS Z) (ADD1 (NEG Z))))))), which again simplifies, using linear arithmetic, to: T. Case 3.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE X Y)))), which again simplifies, using linear arithmetic, to: T. Case 3.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (EQUAL (PLUS (POS Z) (NEG X)) 0) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE X Y)))), which again simplifies, using linear arithmetic, to: T. Case 3.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE X Y)))), which again simplifies, using linear arithmetic, to: T. Case 3.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (EQUAL (PLUS (POS Z) (NEG X)) 0) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE X Y)))), which again simplifies, using linear arithmetic, to: T. Case 3.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))) (NOT (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) Y)))), which again simplifies, using linear arithmetic, to: T. Case 3.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PZDIFFERENCE Z Y) (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) Y)))), which again simplifies, applying SUB1-ADD1, NEG-ZN, and POS-ZN, and unfolding PZDIFFERENCE, DIFFERENCE, and PLUS, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (DIFFERENCE (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)) (PZDIFFERENCE (ZMAX (ZN (POS Z) (ADD1 (NEG Z))) Y) (ZMIN (ZN (POS Z) (ADD1 (NEG Z))) X))))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)) (PZDIFFERENCE (ZMAX (ZN (POS Z) (ADD1 (NEG Z))) Y) (ZMIN (ZN (POS Z) (ADD1 (NEG Z))) X))))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.0 0.1 ] M2-LESSEQP-3 (PROVE-LEMMA M3-LESSP-0 (REWRITE) (IMPLIES (AND (NOT (ZLESSEQP X Y)) (EQUAL (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M1 X Y Z))) (LESSP (M3 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M3 X Y Z)))) WARNING: Note that the linear lemma M3-LESSP-0 is being stored under the term: (M3 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)), which is unusual because M3 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M3-LESSP-0 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, AND, and IMPLIES, to the new conjecture: (IMPLIES (AND (ZLESSP Y X) (EQUAL (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M1 X Y Z))) (LESSP (M3 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M3 X Y Z))), which simplifies, rewriting with POS-ZN and NEG-ZN, and opening up ZLESSP, ZSUB1, ZLESSEQP, TAK0, M1, M3, and ZMIN, to the following 36 new conjectures: Case 36.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Z) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN X X))) (PZDIFFERENCE X (ZMIN X Z)))). However this again simplifies, using linear arithmetic, to: T. Case 35.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Z) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN X (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 34.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Z) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN X X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 33.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Z) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN X (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 32.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Z) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 31.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Z) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 30.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Z) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 29.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Z) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 28.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Z) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN Z X))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, expanding the functions ZLESSP and ZMIN, to four new conjectures: Case 28.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE Z (ZMIN Z X)) (PZDIFFERENCE X Z))), which again simplifies, using linear arithmetic, to: T. Case 28.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE Z (ZMIN Z Z)) (PZDIFFERENCE X Z))), which again simplifies, using linear arithmetic, to: T. Case 28.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE Z (ZMIN Z X)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 28.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE Z (ZMIN Z Z)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 27.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Z) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 26.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Z) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN Z X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, expanding the definitions of ZLESSP and ZMIN, to four new conjectures: Case 26.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE Z (ZMIN Z X)) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 26.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE Z (ZMIN Z Z)) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 26.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE Z (ZMIN Z X)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 26.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE Z (ZMIN Z Z)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 25.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Z) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, rewriting with the lemmas NEG-ZN and POS-ZN, and unfolding ZLESSP and ZMIN, to four new formulas: Case 25.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z))))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 25.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (LESSP (PZDIFFERENCE Z (ZMIN Z Z)) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 25.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))))) (LESSP (PZDIFFERENCE Z (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z))))) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 25.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Z)) (PLUS (NEG Z) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (LESSP (PZDIFFERENCE Z (ZMIN Z Z)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 24.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X (ZN (POS X) (ADD1 (NEG X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN X X))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, applying POS-ZN, NEG-ZN, and SUB1-ADD1, and opening up the definitions of ZLESSP, ZMIN, PLUS, LESSP, and PZDIFFERENCE, to the following three new conjectures: Case 24.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))). However this again simplifies, using linear arithmetic, to: T. Case 24.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS X) (NEG X)) 0)) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))), which again simplifies, using linear arithmetic, to: T. Case 24.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG X))) (PLUS (NEG X) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG X))) (PLUS (NEG X) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) X) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))), which again simplifies, using linear arithmetic, to: T. Case 23.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X (ZN (POS X) (ADD1 (NEG X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN X (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 22.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X (ZN (POS X) (ADD1 (NEG X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN X X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 21.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X (ZN (POS X) (ADD1 (NEG X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN X (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 20.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS X) (ADD1 (NEG X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 19.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS X) (ADD1 (NEG X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 18.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS X) (ADD1 (NEG X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 17.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS X) (ADD1 (NEG X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 16.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z (ZN (POS X) (ADD1 (NEG X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN Z X))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, rewriting with POS-ZN and NEG-ZN, and unfolding the definitions of ZLESSP and ZMIN, to the following four new goals: Case 16.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) X)) (PZDIFFERENCE X Z))). But this again simplifies, using linear arithmetic, to: T. Case 16.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Z)) (PZDIFFERENCE X Z))), which again simplifies, rewriting with POS-ZN, NEG-ZN, and SUB1-ADD1, and unfolding the functions ZLESSP, PLUS, LESSP, ZMIN, and PZDIFFERENCE, to the following three new conjectures: Case 16.3.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))). This again simplifies, using linear arithmetic, to: T. Case 16.3.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (EQUAL (PLUS (POS X) (NEG Z)) 0)) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))), which again simplifies, using linear arithmetic, to: T. Case 16.3.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Z) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))), which again simplifies, applying the lemmas SUB1-ADD1, NEG-ZN, and POS-ZN, and unfolding DIFFERENCE, PLUS, and PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (DIFFERENCE (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))). But this again simplifies, using linear arithmetic, to: T. Case 16.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) X)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 16.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG X))) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Z)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 15.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z (ZN (POS X) (ADD1 (NEG X)))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 14.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z (ZN (POS X) (ADD1 (NEG X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN Z X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 13.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z (ZN (POS X) (ADD1 (NEG X)))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 12.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Y) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN X X))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, expanding ZLESSP, ZMIN, and PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (DIFFERENCE (PLUS (POS Y) (NEG Y)) (PLUS (NEG Y) (POS Y))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))). This again simplifies, using linear arithmetic, to the conjecture: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG Y)) (PLUS (NEG Y) (POS Y))) (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (DIFFERENCE (PLUS (POS Y) (NEG Y)) (PLUS (NEG Y) (POS Y))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))). But this again simplifies, using linear arithmetic, to: T. Case 11.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Y) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN X (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 10.(IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Y) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN X X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (ZLESSP X Y) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN X (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Y) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Y) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Y) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (ZLESSP (ZN (POS Y) (ADD1 (NEG Y))) Y) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Y) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN Z X))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, expanding ZLESSP and ZMIN, to four new goals: Case 4.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE Y (ZMIN Y X)) (PZDIFFERENCE X Z))), which again simplifies, using linear arithmetic, to: T. Case 4.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE Y (ZMIN Y Z)) (PZDIFFERENCE X Z))), which again simplifies, expanding the functions ZLESSP, ZMIN, and PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (DIFFERENCE (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))). However this again simplifies, using linear arithmetic, to: T. Case 4.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE Y (ZMIN Y X)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 4.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE Y (ZMIN Y Z)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Y) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Y) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN Z X))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, expanding the functions ZLESSP and ZMIN, to four new conjectures: Case 2.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE Y (ZMIN Y X)) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE Y (ZMIN Y Z)) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE Y (ZMIN Y X)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE Y (ZMIN Y Z)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (ZLESSP Z Y) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZMIN Z (ZN (POS Z) (ADD1 (NEG Z)))))) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, rewriting with the lemmas NEG-ZN and POS-ZN, and opening up ZLESSP and ZMIN, to four new formulas: Case 1.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZN (POS Z) (ADD1 (NEG Z))))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 1.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (LESSP (PZDIFFERENCE Y (ZMIN Y Z)) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 1.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z))))) (LESSP (PZDIFFERENCE Y (ZMIN Y (ZN (POS Z) (ADD1 (NEG Z))))) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X)))) (NOT (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (PLUS (POS Z) (ADD1 (NEG Z))) (PLUS (NEG Z) (POS Z)))) (LESSP (PZDIFFERENCE Y (ZMIN Y Z)) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.5 0.1 ] M3-LESSP-0 (PROVE-LEMMA M3-LESSP-1 (REWRITE) (IMPLIES (AND (NOT (ZLESSEQP X Y)) (EQUAL (M1 (ZSUB1 X) Y Z) (M1 X Y Z))) (LESSP (M3 (ZSUB1 X) Y Z) (M3 X Y Z)))) WARNING: Note that the linear lemma M3-LESSP-1 is being stored under the term (M3 (ZSUB1 X) Y Z), which is unusual because M3 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M3-LESSP-1 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, AND, and IMPLIES, to the new conjecture: (IMPLIES (AND (ZLESSP Y X) (EQUAL (M1 (ZSUB1 X) Y Z) (M1 X Y Z))) (LESSP (M3 (ZSUB1 X) Y Z) (M3 X Y Z))), which simplifies, rewriting with POS-ZN and NEG-ZN, and opening up the functions ZLESSP, ZSUB1, ZLESSEQP, M1, ZMIN, and M3, to the following two new formulas: Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Z)) (PZDIFFERENCE X (ZMIN X Z)))). This again simplifies, applying POS-ZN, NEG-ZN, and SUB1-ADD1, and unfolding the definitions of ZLESSP, PLUS, LESSP, and ZMIN, to the following six new goals: Case 2.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (PZDIFFERENCE X Z))). This again simplifies, using linear arithmetic, to: T. Case 2.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS X) (NEG Z)) 0)) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (PZDIFFERENCE X Z))), which again simplifies, using linear arithmetic, to: T. Case 2.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Z) (PZDIFFERENCE X Z))), which again simplifies, rewriting with SUB1-ADD1, NEG-ZN, and POS-ZN, and opening up the definitions of DIFFERENCE, PLUS, and PZDIFFERENCE, to the new conjecture: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (DIFFERENCE (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))), which again simplifies, using linear arithmetic, to: T. Case 2.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 2.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS X) (NEG Z)) 0)) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS X) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Z))) (PLUS (NEG X) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Z) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZMIN (ZN (POS X) (ADD1 (NEG X))) Y)) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, applying POS-ZN, NEG-ZN, and SUB1-ADD1, and opening up ZLESSP, PLUS, LESSP, and ZMIN, to the following six new goals: Case 1.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (PZDIFFERENCE X Y))). However this again simplifies, using linear arithmetic, to: T. Case 1.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (EQUAL (PLUS (POS X) (NEG Y)) 0)) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 1.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Y) (PZDIFFERENCE X Y))), which again simplifies, rewriting with SUB1-ADD1, NEG-ZN, and POS-ZN, and expanding the definitions of DIFFERENCE, PLUS, and PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))))) (LESSP (DIFFERENCE (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))) (DIFFERENCE (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))), which again simplifies, using linear arithmetic, to: T. Case 1.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y)))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 1.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (EQUAL (PLUS (POS X) (NEG Y)) 0)) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) (ZN (POS X) (ADD1 (NEG X)))) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 1.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (ADD1 (NEG X))) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))) (NOT (EQUAL (PLUS (POS X) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS X) (NEG Y))) (PLUS (NEG X) (POS Y))))) (LESSP (PZDIFFERENCE (ZN (POS X) (ADD1 (NEG X))) Y) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.1 0.0 ] M3-LESSP-1 (PROVE-LEMMA M3-LESSP-2 (REWRITE) (IMPLIES (AND (NOT (ZLESSEQP X Y)) (EQUAL (M1 (ZSUB1 Y) Z X) (M1 X Y Z))) (LESSP (M3 (ZSUB1 Y) Z X) (M3 X Y Z)))) WARNING: Note that the linear lemma M3-LESSP-2 is being stored under the term (M3 (ZSUB1 Y) Z X), which is unusual because M3 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M3-LESSP-2 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, AND, and IMPLIES, to the new conjecture: (IMPLIES (AND (ZLESSP Y X) (EQUAL (M1 (ZSUB1 Y) Z X) (M1 X Y Z))) (LESSP (M3 (ZSUB1 Y) Z X) (M3 X Y Z))), which simplifies, rewriting with POS-ZN and NEG-ZN, and opening up the functions ZLESSP, ZSUB1, ZLESSEQP, M1, ZMIN, and M3, to the following four new formulas: Case 4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X)) (PZDIFFERENCE X (ZMIN X Z)))). This again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) Z)) (PZDIFFERENCE X (ZMIN X Z)))), which again simplifies, applying POS-ZN, NEG-ZN, and SUB1-ADD1, and opening up the functions ZLESSP, PLUS, LESSP, and ZMIN, to the following six new conjectures: Case 3.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y)))) (PZDIFFERENCE X Z))). However this again simplifies, using linear arithmetic, to: T. Case 3.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y)))) (PZDIFFERENCE X Z))), which again simplifies, using linear arithmetic, to: T. Case 3.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) Z) (PZDIFFERENCE X Z))), which again simplifies, applying SUB1-ADD1, NEG-ZN, and POS-ZN, and opening up DIFFERENCE, PLUS, and PZDIFFERENCE, to: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))))) (LESSP (DIFFERENCE (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))) (DIFFERENCE (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))))), which again simplifies, using linear arithmetic, to: T. Case 3.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z)))) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y)))) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 3.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZN (POS Y) (ADD1 (NEG Y)))) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 3.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS Y) (NEG Z)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Y) (NEG Z))) (PLUS (NEG Y) (POS Z))))) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) Z) (PZDIFFERENCE X X))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) X)) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS Z) (ADD1 (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS Z) (NEG X)) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE (ZN (POS Y) (ADD1 (NEG Y))) (ZMIN (ZN (POS Y) (ADD1 (NEG Y))) Z)) (PZDIFFERENCE X (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.1 0.0 ] M3-LESSP-2 (PROVE-LEMMA M3-LESSP-3 (REWRITE) (IMPLIES (AND (NOT (ZLESSEQP X Y)) (EQUAL (M1 (ZSUB1 Z) X Y) (M1 X Y Z))) (LESSP (M2 (ZSUB1 Z) X Y) (M2 X Y Z)))) WARNING: Note that the linear lemma M3-LESSP-3 is being stored under the term (M2 (ZSUB1 Z) X Y), which is unusual because M2 is a nonrecursive function symbol. WARNING: Note that the proposed lemma M3-LESSP-3 is to be stored as zero type prescription rules, zero compound recognizer rules, one linear rule, and zero replacement rules. This formula can be simplified, using the abbreviations ZLESSEQP, NOT, AND, and IMPLIES, to the new conjecture: (IMPLIES (AND (ZLESSP Y X) (EQUAL (M1 (ZSUB1 Z) X Y) (M1 X Y Z))) (LESSP (M2 (ZSUB1 Z) X Y) (M2 X Y Z))), which simplifies, rewriting with POS-ZN and NEG-ZN, and opening up the functions ZLESSP, ZSUB1, ZLESSEQP, M1, ZMIN, ZMAX, and M2, to the following four new formulas: Case 4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (LESSP (PZDIFFERENCE (ZMAX (ZN (POS Z) (ADD1 (NEG Z))) X) (ZMIN (ZN (POS Z) (ADD1 (NEG Z))) Y)) (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)))). This again simplifies, using linear arithmetic, to: T. Case 3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (NOT (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z)))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PZDIFFERENCE (ZMAX (ZN (POS Z) (ADD1 (NEG Z))) Y) (ZMIN (ZN (POS Z) (ADD1 (NEG Z))) X)) (PZDIFFERENCE (ZMAX X Y) (ZMIN X Z)))), which again simplifies, using linear arithmetic, to: T. Case 2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y))))) (LESSP (PZDIFFERENCE (ZMAX (ZN (POS Z) (ADD1 (NEG Z))) X) (ZMIN (ZN (POS Z) (ADD1 (NEG Z))) Y)) (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)))), which again simplifies, rewriting with the lemmas POS-ZN, NEG-ZN, and SUB1-ADD1, and expanding the functions ZLESSP, PLUS, LESSP, ZMAX, and ZMIN, to 18 new goals: Case 2.18. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (EQUAL (PLUS (POS Z) (NEG X)) 0)) (LESSP (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.17. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.16. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (EQUAL (PLUS (POS Z) (NEG X)) 0)) (LESSP (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.15. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.14. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.13. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.12. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.11. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (EQUAL (PLUS (POS Z) (NEG X)) 0)) (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.10. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (NOT (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z)))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) Y) (PZDIFFERENCE X Y))), which again simplifies, using linear arithmetic, to: T. Case 2.9. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (EQUAL (PLUS (POS Z) (NEG X)) 0)) (LESSP (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE Z Y))), which again simplifies, using linear arithmetic, to: T. Case 2.8. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE Z Y))), which again simplifies, using linear arithmetic, to: T. Case 2.7. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (EQUAL (PLUS (POS Z) (NEG X)) 0)) (LESSP (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE Z Y))), which again simplifies, using linear arithmetic, to: T. Case 2.6. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE X (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE Z Y))), which again simplifies, using linear arithmetic, to: T. Case 2.5. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE Z Y))), which again simplifies, using linear arithmetic, to: T. Case 2.4. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (EQUAL (PLUS (POS Z) (NEG Y)) 0) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) (ZN (POS Z) (ADD1 (NEG Z)))) (PZDIFFERENCE Z Y))), which again simplifies, using linear arithmetic, to: T. Case 2.3. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X)))) (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE Z Y))), which again simplifies, using linear arithmetic, to: T. Case 2.2. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (EQUAL (PLUS (POS Z) (NEG X)) 0)) (LESSP (PZDIFFERENCE X Y) (PZDIFFERENCE Z Y))), which again simplifies, using linear arithmetic, to: T. Case 2.1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))))) (LESSP (PZDIFFERENCE (ZN (POS Z) (ADD1 (NEG Z))) Y) (PZDIFFERENCE Z Y))), which again simplifies, applying SUB1-ADD1, NEG-ZN, and POS-ZN, and unfolding DIFFERENCE, PLUS, and PZDIFFERENCE, to the new formula: (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (NOT (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PLUS (POS X) (NEG Z)) (PLUS (NEG X) (POS Z))) (NOT (EQUAL (PLUS (POS Z) (NEG Y)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y)))) (NOT (EQUAL (PLUS (POS Z) (NEG X)) 0)) (NOT (LESSP (SUB1 (PLUS (POS Z) (NEG X))) (PLUS (NEG Z) (POS X))))) (LESSP (DIFFERENCE (SUB1 (PLUS (POS Z) (NEG Y))) (PLUS (NEG Z) (POS Y))) (DIFFERENCE (PLUS (POS Z) (NEG Y)) (PLUS (NEG Z) (POS Y))))), which again simplifies, using linear arithmetic, to: T. Case 1. (IMPLIES (AND (LESSP (PLUS (POS Y) (NEG X)) (PLUS (NEG Y) (POS X))) (LESSP (PLUS (POS X) (ADD1 (NEG Z))) (PLUS (NEG X) (POS Z))) (LESSP (PLUS (POS Y) (NEG Z)) (PLUS (NEG Y) (POS Z))) (LESSP (PLUS (POS X) (NEG Y)) (PLUS (NEG X) (POS Y)))) (LESSP (PZDIFFERENCE (ZMAX (ZN (POS Z) (ADD1 (NEG Z))) Y) (ZMIN (ZN (POS Z) (ADD1 (NEG Z))) X)) (PZDIFFERENCE (ZMAX X Z) (ZMIN X Y)))), which again simplifies, using linear arithmetic, to: T. Q.E.D. [ 0.0 0.1 0.0 ] M3-LESSP-3 (DISABLE ZLESSP) [ 0.0 0.0 0.0 ] ZLESSP-OFF (DISABLE M1) [ 0.0 0.0 0.0 ] M1-OFF (DISABLE M2) [ 0.0 0.0 0.0 ] M2-OFF (DISABLE M3) [ 0.0 0.0 0.0 ] M3-OFF (DISABLE TAK0) [ 0.0 0.0 0.0 ] TAK0-OFF (DISABLE ZSUB1) [ 0.0 0.0 0.0 ] ZSUB1-OFF (DEFN MAKE-ORDINAL3 (X) (CONS (CONS (ADD1 (CAR X)) 0) (CONS (ADD1 (CADR X)) (FIX (CADDR X))))) From the definition we can conclude that (LISTP (MAKE-ORDINAL3 X)) is a theorem. [ 0.0 0.0 0.0 ] MAKE-ORDINAL3 (PROVE-LEMMA ORDINALP-MAKE-ORDINAL3 NIL (ORDINALP (MAKE-ORDINAL3 X))) This simplifies, opening up the function MAKE-ORDINAL3, to two new formulas: Case 2. (IMPLIES (NOT (NUMBERP (CADDR X))) (ORDINALP (CONS (CONS (ADD1 (CAR X)) 0) (CONS (ADD1 (CADR X)) 0)))), which again simplifies, applying the lemmas CDR-CONS and CAR-CONS, and expanding the definitions of ORD-LESSP, ORDINALP, and LISTP, to: T. Case 1. (IMPLIES (NUMBERP (CADDR X)) (ORDINALP (CONS (CONS (ADD1 (CAR X)) 0) (CONS (ADD1 (CADR X)) (CADDR X))))), which again simplifies, applying CDR-CONS and CAR-CONS, and expanding ORD-LESSP, ORDINALP, and LISTP, to: T. Q.E.D. [ 0.0 0.0 0.0 ] ORDINALP-MAKE-ORDINAL3 (DEFN LEX3 (X Y) (ORD-LESSP (MAKE-ORDINAL3 X) (MAKE-ORDINAL3 Y))) Note that (OR (FALSEP (LEX3 X Y)) (TRUEP (LEX3 X Y))) is a theorem. [ 0.0 0.0 0.0 ] LEX3 (PROVE-LEMMA M-GOES-DOWN-1 NIL (IMPLIES (NOT (ZLESSEQP X Y)) (LEX3 (M (ZSUB1 X) Y Z) (M X Y Z)))) This formula can be simplified, using the abbreviations ZLESSEQP, NOT, IMPLIES, and LEX3, to: (IMPLIES (ZLESSP Y X) (ORD-LESSP (MAKE-ORDINAL3 (M (ZSUB1 X) Y Z)) (MAKE-ORDINAL3 (M X Y Z)))), which simplifies, rewriting with CAR-CONS, CDR-CONS, CONS-EQUAL, ADD1-EQUAL, and SUB1-ADD1, and expanding the functions M, MAKE-ORDINAL3, EQUAL, ORD-LESSP, and LESSP, to the following three new goals: Case 3. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 X) Y Z) (M1 X Y Z))) (NOT (LESSP (M2 (ZSUB1 X) Y Z) (M2 X Y Z)))) (EQUAL (M2 (ZSUB1 X) Y Z) (M2 X Y Z))). This again simplifies, using linear arithmetic, applying the lemma M2-LESSEQP-1, and opening up the definition of ZLESSEQP, to: T. Case 2. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 X) Y Z) (M1 X Y Z))) (NOT (LESSP (M2 (ZSUB1 X) Y Z) (M2 X Y Z)))) (LESSP (M3 (ZSUB1 X) Y Z) (M3 X Y Z))), which again simplifies, using linear arithmetic, applying the lemmas M3-LESSP-1 and M1-LESSEQP-1, and expanding the definition of ZLESSEQP, to: T. Case 1. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 X) Y Z) (M1 X Y Z)))) (EQUAL (M1 (ZSUB1 X) Y Z) (M1 X Y Z))), which again simplifies, using linear arithmetic, applying M1-LESSEQP-1, and expanding the function ZLESSEQP, to: T. Q.E.D. [ 0.0 0.0 0.0 ] M-GOES-DOWN-1 (PROVE-LEMMA M-GOES-DOWN-2 NIL (IMPLIES (NOT (ZLESSEQP X Y)) (LEX3 (M (ZSUB1 Y) Z X) (M X Y Z)))) This formula can be simplified, using the abbreviations ZLESSEQP, NOT, IMPLIES, and LEX3, to: (IMPLIES (ZLESSP Y X) (ORD-LESSP (MAKE-ORDINAL3 (M (ZSUB1 Y) Z X)) (MAKE-ORDINAL3 (M X Y Z)))), which simplifies, rewriting with CAR-CONS, CDR-CONS, CONS-EQUAL, ADD1-EQUAL, and SUB1-ADD1, and expanding the functions M, MAKE-ORDINAL3, EQUAL, ORD-LESSP, and LESSP, to the following three new goals: Case 3. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 Y) Z X) (M1 X Y Z))) (NOT (LESSP (M2 (ZSUB1 Y) Z X) (M2 X Y Z)))) (EQUAL (M2 (ZSUB1 Y) Z X) (M2 X Y Z))). This again simplifies, using linear arithmetic, applying the lemmas M2-LESSEQP-2 and M1-LESSEQP-2, and opening up the definition of ZLESSEQP, to: T. Case 2. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 Y) Z X) (M1 X Y Z))) (NOT (LESSP (M2 (ZSUB1 Y) Z X) (M2 X Y Z)))) (LESSP (M3 (ZSUB1 Y) Z X) (M3 X Y Z))), which again simplifies, using linear arithmetic, applying the lemmas M3-LESSP-2 and M1-LESSEQP-2, and expanding the definition of ZLESSEQP, to: T. Case 1. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 Y) Z X) (M1 X Y Z)))) (EQUAL (M1 (ZSUB1 Y) Z X) (M1 X Y Z))), which again simplifies, using linear arithmetic, applying M1-LESSEQP-2, and expanding the function ZLESSEQP, to: T. Q.E.D. [ 0.0 0.0 0.0 ] M-GOES-DOWN-2 (PROVE-LEMMA M-GOES-DOWN-3 NIL (IMPLIES (NOT (ZLESSEQP X Y)) (LEX3 (M (ZSUB1 Z) X Y) (M X Y Z)))) This formula can be simplified, using the abbreviations ZLESSEQP, NOT, IMPLIES, and LEX3, to: (IMPLIES (ZLESSP Y X) (ORD-LESSP (MAKE-ORDINAL3 (M (ZSUB1 Z) X Y)) (MAKE-ORDINAL3 (M X Y Z)))), which simplifies, rewriting with CAR-CONS, CDR-CONS, CONS-EQUAL, ADD1-EQUAL, and SUB1-ADD1, and expanding the functions M, MAKE-ORDINAL3, EQUAL, ORD-LESSP, and LESSP, to the following three new goals: Case 3. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 Z) X Y) (M1 X Y Z))) (NOT (LESSP (M2 (ZSUB1 Z) X Y) (M2 X Y Z)))) (EQUAL (M2 (ZSUB1 Z) X Y) (M2 X Y Z))). This again simplifies, using linear arithmetic, applying the lemmas M3-LESSP-3 and M1-LESSEQP-3, and opening up the definition of ZLESSEQP, to: T. Case 2. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 Z) X Y) (M1 X Y Z))) (NOT (LESSP (M2 (ZSUB1 Z) X Y) (M2 X Y Z)))) (LESSP (M3 (ZSUB1 Z) X Y) (M3 X Y Z))), which again simplifies, using linear arithmetic, applying the lemmas M3-LESSP-3 and M1-LESSEQP-3, and expanding the definition of ZLESSEQP, to: T. Case 1. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (ZSUB1 Z) X Y) (M1 X Y Z)))) (EQUAL (M1 (ZSUB1 Z) X Y) (M1 X Y Z))), which again simplifies, using linear arithmetic, applying M1-LESSEQP-3, and expanding the function ZLESSEQP, to: T. Q.E.D. [ 0.0 0.0 0.0 ] M-GOES-DOWN-3 (PROVE-LEMMA M-GOES-DOWN-0 NIL (IMPLIES (NOT (ZLESSEQP X Y)) (LEX3 (M (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M X Y Z))) NIL) This formula can be simplified, using the abbreviations ZLESSEQP, NOT, IMPLIES, and LEX3, to the new conjecture: (IMPLIES (ZLESSP Y X) (ORD-LESSP (MAKE-ORDINAL3 (M (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y))) (MAKE-ORDINAL3 (M X Y Z)))), which simplifies, rewriting with CAR-CONS, CDR-CONS, CONS-EQUAL, ADD1-EQUAL, and SUB1-ADD1, and opening up the functions M, MAKE-ORDINAL3, EQUAL, ORD-LESSP, and LESSP, to the following three new formulas: Case 3. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M1 X Y Z))) (NOT (LESSP (M2 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M2 X Y Z)))) (EQUAL (M2 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M2 X Y Z))). This again simplifies, using linear arithmetic, applying M2-LESSEQP-0, and unfolding the definition of ZLESSEQP, to: T. Case 2. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M1 X Y Z))) (NOT (LESSP (M2 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M2 X Y Z)))) (LESSP (M3 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M3 X Y Z))). But this again simplifies, using linear arithmetic, applying M3-LESSP-0 and M1-LESSEQP-0, and opening up ZLESSEQP, to: T. Case 1. (IMPLIES (AND (ZLESSP Y X) (NOT (LESSP (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M1 X Y Z)))) (EQUAL (M1 (TAK0 (ZSUB1 X) Y Z) (TAK0 (ZSUB1 Y) Z X) (TAK0 (ZSUB1 Z) X Y)) (M1 X Y Z))). However this again simplifies, using linear arithmetic, rewriting with M1-LESSEQP-0, and opening up the definition of ZLESSEQP, to: T. Q.E.D. [ 0.0 0.0 0.0 ] M-GOES-DOWN-0