%   ORIGINAL: h4/numRing/num__rewrites_c68
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/numeral/numeral__mult_c0: !n. h4/arithmetic/_2A h4/arithmetic/ZERO n = h4/arithmetic/ZERO
% Assm: h4/numeral/enumeral__mult_c0: !n. h4/arithmetic/_2A h4/arithmetic/ZERO n = h4/arithmetic/ZERO
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/TRUTH: T
% Assm: h4/arithmetic/RIGHT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/MULT__CLAUSES_c0: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/BIT20: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm: h4/arithmetic/BIT10: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm: h4/arithmetic/MULT__CLAUSES_c4: !n m. h4/arithmetic/_2A (h4/num/SUC m) n = h4/arithmetic/_2B (h4/arithmetic/_2A m n) n
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/numeral/iDUB0: !x. h4/numeral/iDUB x = h4/arithmetic/_2B x x
% Assm: h4/numeral/iZ0: !x. h4/numeral/iZ x = x
% Assm: h4/arithmetic/ALT__ZERO: h4/arithmetic/ZERO = h4/num/0
% Assm: h4/numRing/num__semi__ring: h4/semi__ring/is__semi__ring (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__interp__sp__def: h4/numRing/num__interp__sp = h4/canonical/interp__sp (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__canonical__sum__simplify__def: h4/numRing/num__canonical__sum__simplify = h4/canonical/canonical__sum__simplify (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__monom__insert__def: h4/numRing/num__monom__insert = h4/canonical/monom__insert (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__canonical__sum__prod__def: h4/numRing/num__canonical__sum__prod = h4/canonical/canonical__sum__prod (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__canonical__sum__scalar__def: h4/numRing/num__canonical__sum__scalar = h4/canonical/canonical__sum__scalar (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__interp__cs__def: h4/numRing/num__interp__cs = h4/canonical/interp__cs (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__spolynom__simplify__def: h4/numRing/num__spolynom__simplify = h4/canonical/spolynom__simplify (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__canonical__sum__scalar3__def: h4/numRing/num__canonical__sum__scalar3 = h4/canonical/canonical__sum__scalar3 (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__spolynom__normalize__def: h4/numRing/num__spolynom__normalize = h4/canonical/spolynom__normalize (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__ics__aux__def: h4/numRing/num__ics__aux = h4/canonical/ics__aux (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__interp__vl__def: h4/numRing/num__interp__vl = h4/canonical/interp__vl (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__ivl__aux__def: h4/numRing/num__ivl__aux = h4/canonical/ivl__aux (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__varlist__insert__def: h4/numRing/num__varlist__insert = h4/canonical/varlist__insert (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__interp__m__def: h4/numRing/num__interp__m = h4/canonical/interp__m (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__canonical__sum__scalar2__def: h4/numRing/num__canonical__sum__scalar2 = h4/canonical/canonical__sum__scalar2 (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__canonical__sum__merge__def: h4/numRing/num__canonical__sum__merge = h4/canonical/canonical__sum__merge (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__ring__thms_c0: h4/semi__ring/is__semi__ring (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numeral/internal__mult__def: h4/numeral/internal__mult = h4/arithmetic/_2A
% Assm: h4/arithmetic/DIVMOD__DEF: h4/arithmetic/DIVMOD = h4/relation/WFREC (h4/prim__rec/measure (h4/combin/o h4/pair/FST h4/pair/SND)) (\f. h4/pair/UNCURRY (\a. h4/pair/UNCURRY (\m n. h4/bool/COND (n = h4/num/0) (h4/pair/_2C h4/num/0 h4/num/0) (h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (\q. f (h4/pair/_2C (h4/arithmetic/_2B a q) (h4/pair/_2C (h4/arithmetic/_2D m (h4/arithmetic/_2A n q)) n))) (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n))))))))
% Assm: h4/numeral/divmod__POS: !n m a. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIVMOD (h4/pair/_2C a (h4/pair/_2C m n)) = h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (\q. h4/arithmetic/DIVMOD (h4/pair/_2C (h4/arithmetic/_2B a q) (h4/pair/_2C (h4/arithmetic/_2D m (h4/arithmetic/_2A n q)) n))) (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n))))
% Assm: h4/numeral/numeral__mult_c1: !n. h4/arithmetic/_2A n h4/arithmetic/ZERO = h4/arithmetic/ZERO
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/numeral/texp__help__thm: !n a. h4/numeral/texp__help n a = h4/arithmetic/_2A (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/arithmetic/_2B n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/semi__ring/is__semi__ring__def: !r. h4/semi__ring/is__semi__ring r <=> (!n m. h4/semi__ring/semi__ring__SRP r n m = h4/semi__ring/semi__ring__SRP r m n) /\ (!n m p. h4/semi__ring/semi__ring__SRP r n (h4/semi__ring/semi__ring__SRP r m p) = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r n m) p) /\ (!n m. h4/semi__ring/semi__ring__SRM r n m = h4/semi__ring/semi__ring__SRM r m n) /\ (!n m p. h4/semi__ring/semi__ring__SRM r n (h4/semi__ring/semi__ring__SRM r m p) = h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r n m) p) /\ (!n. h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SR0 r) n = n) /\ (!n. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SR1 r) n = n) /\ (!n. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SR0 r) n = h4/semi__ring/semi__ring__SR0 r) /\ (!n m p. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRP r n m) p = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRM r n p) (h4/semi__ring/semi__ring__SRM r m p))
% Assm: h4/semi__ring/semi__ring__accessors_c3: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/semi__ring/semi__ring__accessors_c0: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/semi__ring/semi__ring__accessors_c1: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring0 a a0 f f0) = a0
% Assm: h4/semi__ring/semi__ring__accessors_c2: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm: h4/arithmetic/MULT__COMM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/arithmetic/MULT__ASSOC: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm: h4/arithmetic/MULT__0: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/arithmetic/MULT__LEFT__1: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/arithmetic/SUB__LEFT__LESS__EQ: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/arithmetic/MULT__EQ__0: !n m. h4/arithmetic/_2A m n = h4/num/0 <=> m = h4/num/0 \/ n = h4/num/0
% Assm: h4/arithmetic/LESS__LESS__EQ__TRANS: !p n m. h4/prim__rec/_3C m n /\ h4/arithmetic/_3C_3D n p ==> h4/prim__rec/_3C m p
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/arithmetic/LESS__ADD__NONZERO: !n m. ~(n = h4/num/0) ==> h4/prim__rec/_3C m (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/arithmetic/findq__eq__0: !n m a. h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/num/0 <=> a = h4/num/0
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/prim__rec/measure__thm: !y x f. h4/prim__rec/measure f x y <=> h4/prim__rec/_3C (f x) (f y)
% Assm: h4/prim__rec/LESS__REFL: !n. ~h4/prim__rec/_3C n n
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/relation/RESTRICT__DEF: !x f R. h4/relation/RESTRICT f R x = (\y. h4/bool/COND (R y x) (f y) h4/bool/ARB)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/arithmetic/NOT__ZERO__LT__ZERO: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/numeral/texp__help__def_c1: !n acc. h4/numeral/texp__help (h4/num/SUC n) acc = h4/numeral/texp__help n (h4/arithmetic/BIT1 acc)
% Assm: h4/arithmetic/TIMES2: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm: h4/numeral/texp__help__def_c0: !acc. h4/numeral/texp__help h4/num/0 acc = h4/arithmetic/BIT2 acc
% Assm: h4/arithmetic/LEFT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm: h4/arithmetic/MULT__CLAUSES_c5: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__CLAUSES_c2: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/TWO: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/arithmetic/EXP0_c1: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm: h4/arithmetic/EXP0_c0: !m. h4/arithmetic/EXP m h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Goal: !n. h4/arithmetic/_2A h4/arithmetic/ZERO n = h4/arithmetic/ZERO
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_numerals_numeralu_u_multu_c0]: !n. happ (happ h4/arithmetic/_2A h4/arithmetic/ZERO) n = h4/arithmetic/ZERO
% Assm [h4s_numerals_enumeralu_u_multu_c0]: !n. happ (happ h4/arithmetic/_2A h4/arithmetic/ZERO) n = h4/arithmetic/ZERO
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB]: !p n m. happ (happ h4/arithmetic/_2A (happ (happ h4/arithmetic/_2B m) n)) p = happ (happ h4/arithmetic/_2B (happ (happ h4/arithmetic/_2A m) p)) (happ (happ h4/arithmetic/_2A n) p)
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. happ (happ h4/arithmetic/_2B m) (happ (happ h4/arithmetic/_2B n) p) = happ (happ h4/arithmetic/_2B (happ (happ h4/arithmetic/_2B m) n)) p
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. happ (happ h4/arithmetic/_2B m) n = happ (happ h4/arithmetic/_2B n) m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. happ (happ h4/arithmetic/_2B m) (h4/num/SUC n) = h4/num/SUC (happ (happ h4/arithmetic/_2B m) n)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. happ (happ h4/arithmetic/_2B m) h4/num/0 = m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c0]: !m. happ (happ h4/arithmetic/_2A h4/num/0) m = h4/num/0
% Assm [h4s_arithmetics_BIT20]: !n. h4/arithmetic/BIT2 n = happ (happ h4/arithmetic/_2B n) (happ (happ h4/arithmetic/_2B n) (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm [h4s_arithmetics_BIT10]: !n. h4/arithmetic/BIT1 n = happ (happ h4/arithmetic/_2B n) (happ (happ h4/arithmetic/_2B n) (h4/num/SUC h4/num/0))
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c4]: !n m. happ (happ h4/arithmetic/_2A (h4/num/SUC m)) n = happ (happ h4/arithmetic/_2B (happ (happ h4/arithmetic/_2A m) n)) n
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. happ (happ h4/arithmetic/_2A m) h4/num/0 = h4/num/0
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_numerals_iDUB0]: !x. h4/numeral/iDUB x = happ (happ h4/arithmetic/_2B x) x
% Assm [h4s_numerals_iZ0]: !x. h4/numeral/iZ x = x
% Assm [h4s_arithmetics_ALTu_u_ZERO]: h4/arithmetic/ZERO = h4/num/0
% Assm [h4s_numRings_numu_u_semiu_u_ring]: h4/semi__ring/is__semi__ring (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_interpu_u_spu_u_def]: h4/numRing/num__interp__sp = h4/canonical/interp__sp (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_simplifyu_u_def]: h4/numRing/num__canonical__sum__simplify = h4/canonical/canonical__sum__simplify (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_monomu_u_insertu_u_def]: h4/numRing/num__monom__insert = h4/canonical/monom__insert (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_produ_u_def]: h4/numRing/num__canonical__sum__prod = h4/canonical/canonical__sum__prod (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_scalaru_u_def]: h4/numRing/num__canonical__sum__scalar = h4/canonical/canonical__sum__scalar (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_interpu_u_csu_u_def]: h4/numRing/num__interp__cs = h4/canonical/interp__cs (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_spolynomu_u_simplifyu_u_def]: h4/numRing/num__spolynom__simplify = h4/canonical/spolynom__simplify (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_scalar3u_u_def]: h4/numRing/num__canonical__sum__scalar3 = h4/canonical/canonical__sum__scalar3 (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_spolynomu_u_normalizeu_u_def]: h4/numRing/num__spolynom__normalize = h4/canonical/spolynom__normalize (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_icsu_u_auxu_u_def]: h4/numRing/num__ics__aux = h4/canonical/ics__aux (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_interpu_u_vlu_u_def]: h4/numRing/num__interp__vl = h4/canonical/interp__vl (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_ivlu_u_auxu_u_def]: h4/numRing/num__ivl__aux = h4/canonical/ivl__aux (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_varlistu_u_insertu_u_def]: h4/numRing/num__varlist__insert = h4/canonical/varlist__insert (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_interpu_u_mu_u_def]: h4/numRing/num__interp__m = h4/canonical/interp__m (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_scalar2u_u_def]: h4/numRing/num__canonical__sum__scalar2 = h4/canonical/canonical__sum__scalar2 (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_mergeu_u_def]: h4/numRing/num__canonical__sum__merge = h4/canonical/canonical__sum__merge (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c0]: h4/semi__ring/is__semi__ring (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numerals_internalu_u_multu_u_def]: h4/numeral/internal__mult = h4/arithmetic/_2A
% Assm [h4s_arithmetics_DIVMODu_u_DEF]: !_4. (!f a m n q. happ (happ (happ (happ (happ _4 f) a) m) n) q = happ f (h4/pair/_2C (happ (happ h4/arithmetic/_2B a) q) (h4/pair/_2C (h4/arithmetic/_2D m (happ (happ h4/arithmetic/_2A n) q)) n))) ==> (!_3. (!f a m n. ?v. (v <=> n = h4/num/0) /\ happ (happ (happ (happ _3 f) a) m) n = h4/bool/COND v (h4/pair/_2C h4/num/0 h4/num/0) (h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (happ (happ (happ (happ _4 f) a) m) n) (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n)))))) ==> (!_2. (!f a m. happ (happ (happ _2 f) a) m = happ (happ (happ _3 f) a) m) ==> (!_1. (!f a. happ (happ _1 f) a = h4/pair/UNCURRY (happ (happ _2 f) a)) ==> (!_0. (!f. happ _0 f = h4/pair/UNCURRY (happ _1 f)) ==> h4/arithmetic/DIVMOD = h4/relation/WFREC (h4/prim__rec/measure (h4/combin/o h4/pair/FST h4/pair/SND)) _0))))
% Assm [h4s_numerals_divmodu_u_POS]: !_0. (!a m n q. happ (happ (happ (happ _0 a) m) n) q = happ h4/arithmetic/DIVMOD (h4/pair/_2C (happ (happ h4/arithmetic/_2B a) q) (h4/pair/_2C (h4/arithmetic/_2D m (happ (happ h4/arithmetic/_2A n) q)) n))) ==> (!n m a. h4/prim__rec/_3C h4/num/0 n ==> happ h4/arithmetic/DIVMOD (h4/pair/_2C a (h4/pair/_2C m n)) = h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (happ (happ (happ _0 a) m) n) (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n)))))
% Assm [h4s_numerals_numeralu_u_multu_c1]: !n. happ (happ h4/arithmetic/_2A n) h4/arithmetic/ZERO = h4/arithmetic/ZERO
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. happ (happ h4/arithmetic/_2B m) n = happ (happ h4/arithmetic/_2B n) m
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_numerals_texpu_u_helpu_u_thm]: !n a. h4/numeral/texp__help n a = happ (happ h4/arithmetic/_2A (happ (happ h4/arithmetic/_2B a) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (happ (happ h4/arithmetic/_2B n) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_semiu_u_rings_isu_u_semiu_u_ringu_u_def]: !r. h4/semi__ring/is__semi__ring r <=> (!n m. happ (happ (h4/semi__ring/semi__ring__SRP r) n) m = happ (happ (h4/semi__ring/semi__ring__SRP r) m) n) /\ (!n m p. happ (happ (h4/semi__ring/semi__ring__SRP r) n) (happ (happ (h4/semi__ring/semi__ring__SRP r) m) p) = happ (happ (h4/semi__ring/semi__ring__SRP r) (happ (happ (h4/semi__ring/semi__ring__SRP r) n) m)) p) /\ (!n m. happ (happ (h4/semi__ring/semi__ring__SRM r) n) m = happ (happ (h4/semi__ring/semi__ring__SRM r) m) n) /\ (!n m p. happ (happ (h4/semi__ring/semi__ring__SRM r) n) (happ (happ (h4/semi__ring/semi__ring__SRM r) m) p) = happ (happ (h4/semi__ring/semi__ring__SRM r) (happ (happ (h4/semi__ring/semi__ring__SRM r) n) m)) p) /\ (!n. happ (happ (h4/semi__ring/semi__ring__SRP r) (h4/semi__ring/semi__ring__SR0 r)) n = n) /\ (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) (h4/semi__ring/semi__ring__SR1 r)) n = n) /\ (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) (h4/semi__ring/semi__ring__SR0 r)) n = h4/semi__ring/semi__ring__SR0 r) /\ (!n m p. happ (happ (h4/semi__ring/semi__ring__SRM r) (happ (happ (h4/semi__ring/semi__ring__SRP r) n) m)) p = happ (happ (h4/semi__ring/semi__ring__SRP r) (happ (happ (h4/semi__ring/semi__ring__SRM r) n) p)) (happ (happ (h4/semi__ring/semi__ring__SRM r) m) p))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c1]: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring0 a a0 f f0) = a0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm [h4s_arithmetics_MULTu_u_COMM]: !n m. happ (happ h4/arithmetic/_2A m) n = happ (happ h4/arithmetic/_2A n) m
% Assm [h4s_arithmetics_MULTu_u_ASSOC]: !p n m. happ (happ h4/arithmetic/_2A m) (happ (happ h4/arithmetic/_2A n) p) = happ (happ h4/arithmetic/_2A (happ (happ h4/arithmetic/_2A m) n)) p
% Assm [h4s_arithmetics_MULTu_u_0]: !m. happ (happ h4/arithmetic/_2A m) h4/num/0 = h4/num/0
% Assm [h4s_arithmetics_ADDu_u_0]: !m. happ (happ h4/arithmetic/_2B m) h4/num/0 = m
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. happ (happ h4/arithmetic/_2B m) p = happ (happ h4/arithmetic/_2B n) p <=> m = n
% Assm [h4s_arithmetics_MULTu_u_LEFTu_u_1]: !m. happ (happ h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) m = m
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (happ (happ h4/arithmetic/_2B m) p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_arithmetics_MULTu_u_EQu_u_0]: !n m. happ (happ h4/arithmetic/_2A m) n = h4/num/0 <=> m = h4/num/0 \/ n = h4/num/0
% Assm [h4s_arithmetics_LESSu_u_LESSu_u_EQu_u_TRANS]: !p n m. h4/prim__rec/_3C m n /\ h4/arithmetic/_3C_3D n p ==> h4/prim__rec/_3C m p
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_arithmetics_LESSu_u_ADDu_u_NONZERO]: !n m. ~(n = h4/num/0) ==> h4/prim__rec/_3C m (happ (happ h4/arithmetic/_2B m) n)
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_arithmetics_findqu_u_equ_u_0]: !n m a. h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/num/0 <=> a = h4/num/0
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_primu_u_recs_measureu_u_thm]: !y x f. happ (happ (h4/prim__rec/measure f) x) y <=> h4/prim__rec/_3C (happ f x) (happ f y)
% Assm [h4s_primu_u_recs_LESSu_u_REFL]: !n. ~h4/prim__rec/_3C n n
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_pairs_SND0]: !y x. happ h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_relations_RESTRICTu_u_DEF]: !x f R x'. happ (h4/relation/RESTRICT f R x) x' = h4/bool/COND (happ (happ R x') x) (happ f x') h4/bool/ARB
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO]: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_numerals_texpu_u_helpu_u_defu_c1]: !n acc. h4/numeral/texp__help (h4/num/SUC n) acc = h4/numeral/texp__help n (h4/arithmetic/BIT1 acc)
% Assm [h4s_arithmetics_TIMES2]: !n. happ (happ h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n = happ (happ h4/arithmetic/_2B n) n
% Assm [h4s_numerals_texpu_u_helpu_u_defu_c0]: !acc. h4/numeral/texp__help h4/num/0 acc = h4/arithmetic/BIT2 acc
% Assm [h4s_arithmetics_LEFTu_u_ADDu_u_DISTRIB]: !p n m. happ (happ h4/arithmetic/_2A p) (happ (happ h4/arithmetic/_2B m) n) = happ (happ h4/arithmetic/_2B (happ (happ h4/arithmetic/_2A p) m)) (happ (happ h4/arithmetic/_2A p) n)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c5]: !n m. happ (happ h4/arithmetic/_2A m) (h4/num/SUC n) = happ (happ h4/arithmetic/_2B m) (happ (happ h4/arithmetic/_2A m) n)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. happ (happ h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) m = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c2]: !n m. happ (happ h4/arithmetic/_2B (h4/num/SUC m)) n = h4/num/SUC (happ (happ h4/arithmetic/_2B m) n)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. happ (happ h4/arithmetic/_2B h4/num/0) m = m
% Assm [h4s_arithmetics_TWO]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_arithmetics_EXP0u_c1]: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = happ (happ h4/arithmetic/_2A m) (h4/arithmetic/EXP m n)
% Assm [h4s_arithmetics_EXP0u_c0]: !m. h4/arithmetic/EXP m h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Goal: !n. happ (happ h4/arithmetic/_2A h4/arithmetic/ZERO) n = h4/arithmetic/ZERO
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1375358,TV_Q1375354]: ![V_f, V_g]: (![V_x]: s(TV_Q1375354,happ(s(t_fun(TV_Q1375358,TV_Q1375354),V_f),s(TV_Q1375358,V_x))) = s(TV_Q1375354,happ(s(t_fun(TV_Q1375358,TV_Q1375354),V_g),s(TV_Q1375358,V_x))) => s(t_fun(TV_Q1375358,TV_Q1375354),V_f) = s(t_fun(TV_Q1375358,TV_Q1375354),V_g))).
fof(ah4s_numerals_numeralu_u_multu_c0, axiom, ![V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero)).
fof(ah4s_numerals_enumeralu_u_multu_c0, axiom, ![V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero)).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_p))))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_BIT20, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_arithmetics_BIT10, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c4, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_numerals_iDUB0, axiom, ![V_x]: s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_x)))).
fof(ah4s_numerals_iZ0, axiom, ![V_x]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
fof(ah4s_arithmetics_ALTu_u_ZERO, axiom, s(t_h4s_nums_num,h4s_arithmetics_zero) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_numRings_numu_u_semiu_u_ring, axiom, p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a))))))).
fof(ah4s_numRings_numu_u_interpu_u_spu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_sp) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_nums_num)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_canonicalu_u_sumu_u_simplifyu_u_def, axiom, s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),h4s_numrings_numu_u_canonicalu_u_sumu_u_simplify) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),h4s_canonicals_canonicalu_u_sumu_u_simplify(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_monomu_u_insertu_u_def, axiom, s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_monomu_u_insert) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_canonicals_monomu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_canonicalu_u_sumu_u_produ_u_def, axiom, s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_prod) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_canonicals_canonicalu_u_sumu_u_prod(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_canonicalu_u_sumu_u_scalaru_u_def, axiom, s(t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_canonicals_canonicalu_u_sumu_u_scalar(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_interpu_u_csu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_cs) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num)),h4s_canonicals_interpu_u_cs(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_spolynomu_u_simplifyu_u_def, axiom, s(t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),h4s_numrings_numu_u_spolynomu_u_simplify) = s(t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),h4s_canonicals_spolynomu_u_simplify(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_canonicalu_u_sumu_u_scalar3u_u_def, axiom, s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar3) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_canonicals_canonicalu_u_sumu_u_scalar3(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_spolynomu_u_normalizeu_u_def, axiom, s(t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),h4s_numrings_numu_u_spolynomu_u_normalize) = s(t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),h4s_canonicals_spolynomu_u_normalize(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_icsu_u_auxu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num))),h4s_numrings_numu_u_icsu_u_aux) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num))),h4s_canonicals_icsu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_interpu_u_vlu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_vl) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num)),h4s_canonicals_interpu_u_vl(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_ivlu_u_auxu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_numrings_numu_u_ivlu_u_aux) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_varlistu_u_insertu_u_def, axiom, s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_varlistu_u_insert) = s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_canonicals_varlistu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_interpu_u_mu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_numrings_numu_u_interpu_u_m) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_canonicals_interpu_u_m(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_canonicalu_u_sumu_u_scalar2u_u_def, axiom, s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar2) = s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_canonicals_canonicalu_u_sumu_u_scalar2(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_canonicalu_u_sumu_u_mergeu_u_def, axiom, s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_merge) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_canonicals_canonicalu_u_sumu_u_merge(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c0, axiom, p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a))))))).
fof(ah4s_numerals_internalu_u_multu_u_def, axiom, s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_numerals_internalu_u_mult) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)).
fof(ah4s_arithmetics_DIVMODu_u_DEF, axiom, ![V_uu_4]: (![V_f, V_a, V_m, V_n, V_q]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_q))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_q))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_q))))),s(t_h4s_nums_num,V_n))))))) => ![V_uu_3]: (![V_f, V_a, V_m, V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_m))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_findq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))))))))))) => ![V_uu_2]: (![V_f, V_a, V_m]: s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))) = s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))) => ![V_uu_1]: (![V_f, V_a]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))))) => ![V_uu_0]: (![V_f]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_divmod) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)),h4s_primu_u_recs_measure(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_combins_o(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num),h4s_pairs_fst),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_snd))))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0))))))))).
fof(ah4s_numerals_divmodu_u_POS, axiom, ![V_uu_0]: (![V_a, V_m, V_n, V_q]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_q))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_divmod),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_q))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_q))))),s(t_h4s_nums_num,V_n))))))) => ![V_n, V_m, V_a]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) => s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_divmod),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_m))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_findq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))))))))))).
fof(ah4s_numerals_numeralu_u_multu_c1, axiom, ![V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_h4s_nums_num,h4s_arithmetics_zero)).
fof(ah4s_arithmetics_ADDu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_numerals_texpu_u_helpu_u_thm, axiom, ![V_n, V_a]: s(t_h4s_nums_num,h4s_numerals_texpu_u_help(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_a))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
fof(ah4s_semiu_u_rings_isu_u_semiu_u_ringu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) <=> (![V_n, V_m]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_n))) & (![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) & (![V_n, V_m]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_n))) & (![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) & (![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n) & (![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n) & (![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))) & ![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_p))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p)))))))))))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a0)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_arithmetics_MULTu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_MULTu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_MULTu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_arithmetics_MULTu_u_LEFTu_u_1, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ONE, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_MULTu_u_EQu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_arithmetics_LESSu_u_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_bools_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t)).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_arithmetics_LESSu_u_ADDu_u_NONZERO, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_arithmetics_findqu_u_equ_u_0, axiom, ![V_n, V_m, V_a]: (s(t_h4s_nums_num,h4s_arithmetics_findq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_h4s_nums_num,V_a) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_primu_u_recs_measureu_u_thm, axiom, ![TV_u_27a]: ![V_y, V_x, V_f]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_primu_u_recs_measure(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_y)))))).
fof(ah4s_primu_u_recs_LESSu_u_REFL, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_primu_u_recs_WFu_u_measure, axiom, ![TV_u_27a]: ![V_m]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_primu_u_recs_measure(s(t_fun(TV_u_27a,t_h4s_nums_num),V_m))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_relations_RESTRICTu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_R, V_xi_]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_xi_))),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_))),s(TV_u_27b,h4s_bools_arb)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO, axiom, ![V_n]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_numerals_texpu_u_helpu_u_defu_c1, axiom, ![V_n, V_acc]: s(t_h4s_nums_num,h4s_numerals_texpu_u_help(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_acc))) = s(t_h4s_nums_num,h4s_numerals_texpu_u_help(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_acc)))))).
fof(ah4s_arithmetics_TIMES2, axiom, ![V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_texpu_u_helpu_u_defu_c0, axiom, ![V_acc]: s(t_h4s_nums_num,h4s_numerals_texpu_u_help(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_acc))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_acc)))).
fof(ah4s_arithmetics_LEFTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c5, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_TWO, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_arithmetics_EXP0u_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_EXP0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ch4s_numRings_numu_u_rewritesu_c68, conjecture, ![V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero)).
