%   ORIGINAL: h4/rat/RAT__OF__NUM_c1
% 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/rat/rat__of__num__def_c2: !n. h4/rat/rat__of__num (h4/num/SUC (h4/num/SUC n)) = h4/rat/rat__add (h4/rat/rat__of__num (h4/num/SUC n)) h4/rat/rat__1
% Assm: h4/rat/RAT__OF__NUM__CALCULATE: !n1. h4/rat/rat__of__num n1 = h4/rat/abs__rat (h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num n1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))
% Assm: h4/rat/rat__of__num__primitive__def: h4/rat/rat__of__num = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!n. R (h4/num/SUC n) (h4/num/SUC (h4/num/SUC n))))) (\rat__of__num a. h4/arithmetic/num__CASE a (h4/combin/I h4/rat/rat__0) (\v. h4/arithmetic/num__CASE v (h4/combin/I h4/rat/rat__1) (\n. h4/combin/I (h4/rat/rat__add (rat__of__num (h4/num/SUC n)) h4/rat/rat__1))))
% Assm: h4/bool/TRUTH: T
% Assm: h4/rat/rat__of__num__def__compute_c3: !n. h4/rat/rat__of__num (h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n))) = h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n))) h4/rat/rat__1
% Assm: h4/rat/rat__of__num__def__compute_c2: !n. h4/rat/rat__of__num (h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) = h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) h4/rat/rat__1
% Assm: h4/rat/rat__1__def: h4/rat/rat__1 = h4/rat/abs__rat h4/frac/frac__1
% Assm: h4/rat/rat__of__num__def_c1: h4/rat/rat__of__num (h4/num/SUC h4/num/0) = h4/rat/rat__1
% Assm: h4/rat/rat__of__num__def__compute_c1: h4/rat/rat__of__num (h4/num/SUC h4/num/0) = h4/rat/rat__1
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% 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/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% 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__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/num__case__def_c0: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm: h4/arithmetic/num__case__def_c1: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = f n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/arithmetic/SUC__ELIM__NUMERALS: !g f. (!n. g (h4/num/SUC n) = f n (h4/num/SUC n)) <=> (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm: h4/rat/RAT__ADD__RINV: !a. h4/rat/rat__add a (h4/rat/rat__ainv a) = h4/rat/rat__of__num h4/num/0
% Assm: h4/rat/RAT__ADD__RID: !a. h4/rat/rat__add a (h4/rat/rat__of__num h4/num/0) = a
% Assm: h4/rat/RAT__ADD__COMM: !b a. h4/rat/rat__add a b = h4/rat/rat__add b a
% Assm: h4/rat/RAT__ADD__LINV: !a. h4/rat/rat__add (h4/rat/rat__ainv a) a = h4/rat/rat__of__num h4/num/0
% Assm: h4/frac/NMR: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__nmr (h4/frac/abs__frac (h4/pair/_2C a b)) = a
% Assm: h4/rat/RAT__ADD__LID: !a. h4/rat/rat__add (h4/rat/rat__of__num h4/num/0) a = a
% Assm: h4/rat/RAT__LES0__LES0__ADD: !r2 r1. h4/rat/rat__les r1 (h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__les r2 (h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__les (h4/rat/rat__add r1 r2) (h4/rat/rat__of__num h4/num/0)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/rat/RAT__ADD__CALCULATE: !f2 f1. h4/rat/rat__add (h4/rat/abs__rat f1) (h4/rat/abs__rat f2) = h4/rat/abs__rat (h4/frac/frac__add f1 f2)
% Assm: h4/integer/INT__LT__REDUCE_c0: !n. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) <=> T
% Assm: h4/frac/DNM: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__dnm (h4/frac/abs__frac (h4/pair/_2C a b)) = b
% Assm: h4/frac/frac__add__def: !f2 f1. h4/frac/frac__add f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__add (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2)) (h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1))) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% Assm: h4/frac/frac__0__def: h4/frac/frac__0 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/frac/frac__1__def: h4/frac/frac__1 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/integer/INT: !n. h4/integer/int__of__num (h4/num/SUC n) = h4/integer/int__add (h4/integer/int__of__num n) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/integer/INT__MUL: !n m. h4/integer/int__mul (h4/integer/int__of__num m) (h4/integer/int__of__num n) = h4/integer/int__of__num (h4/arithmetic/_2A m n)
% Assm: h4/rat/rat__of__num__ind: !P. P h4/num/0 /\ P (h4/num/SUC h4/num/0) /\ (!n. P (h4/num/SUC n) ==> P (h4/num/SUC (h4/num/SUC n))) ==> (!v. P v)
% Assm: h4/rat/rat__0__def: h4/rat/rat__0 = h4/rat/abs__rat h4/frac/frac__0
% Assm: h4/arithmetic/MULT__RIGHT__1: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/int__arith/INT__NUM__FORALL: !P. (!n. P (h4/integer/int__of__num n)) <=> (!x. h4/integer/int__le (h4/integer/int__of__num h4/num/0) x ==> P x)
% Assm: h4/integer/INT__MUL__LID: !x. h4/integer/int__mul (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm: h4/numeral/numeral__distrib_c13: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% 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/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/rat/rat__00: h4/rat/rat__of__num h4/num/0 = h4/rat/abs__rat h4/frac/frac__0
% Assm: h4/rat/rat__add__def: !r2 r1. h4/rat/rat__add r1 r2 = h4/rat/abs__rat (h4/frac/frac__add (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm: h4/rat/RAT__ADD__CONG_c1: !y x. h4/rat/abs__rat (h4/frac/frac__add x (h4/rat/rep__rat (h4/rat/abs__rat y))) = h4/rat/abs__rat (h4/frac/frac__add x y)
% Assm: h4/frac/FRAC__ADD__RID: !a. h4/frac/frac__add a h4/frac/frac__0 = a
% Assm: h4/rat/rat__QUOTIENT: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm: h4/quotient/QUOTIENT__def: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. abs (rep a) = a) /\ (!a. R (rep a) (rep a)) /\ (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/frac/FRAC__DNMPOS: !f. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/frac/frac__dnm f)
% Assm: h4/intExtension/INT__MUL__POS__SIGN: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) a ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul a b)
% Assm: h4/frac/FRAC__ADD__CALCULATE: !b2 b1 a2 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b2 ==> h4/frac/frac__add (h4/frac/abs__frac (h4/pair/_2C a1 b1)) (h4/frac/abs__frac (h4/pair/_2C a2 b2)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__add (h4/integer/int__mul a1 b2) (h4/integer/int__mul a2 b1)) (h4/integer/int__mul b1 b2))
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/frac/FRAC: !f. h4/frac/abs__frac (h4/pair/_2C (h4/frac/frac__nmr f) (h4/frac/frac__dnm f)) = f
% Assm: h4/intExtension/INT__LT__ADD__NEG: !y x. h4/integer/int__lt x (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__lt y (h4/integer/int__of__num h4/num/0) ==> h4/integer/int__lt (h4/integer/int__add x y) (h4/integer/int__of__num h4/num/0)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/rat/RAT__LES0__NMR: !r1. h4/rat/rat__les r1 (h4/rat/rat__of__num h4/num/0) <=> h4/integer/int__lt (h4/rat/rat__nmr r1) (h4/integer/int__of__num h4/num/0)
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/rat/rat__nmr__def: !r. h4/rat/rat__nmr r = h4/frac/frac__nmr (h4/rat/rep__rat r)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/rat/RAT: !r. h4/rat/abs__rat (h4/rat/rep__rat r) = r
% Assm: h4/integer/INT__MUL__SIGN__CASES_c1: !q p. h4/integer/int__lt (h4/integer/int__mul p q) (h4/integer/int__of__num h4/num/0) <=> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) p /\ h4/integer/int__lt q (h4/integer/int__of__num h4/num/0) \/ h4/integer/int__lt p (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__lt (h4/integer/int__of__num h4/num/0) q
% Assm: h4/integer/INT__MUL__SIGN__CASES_c0: !q p. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul p q) <=> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) p /\ h4/integer/int__lt (h4/integer/int__of__num h4/num/0) q \/ h4/integer/int__lt p (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__lt q (h4/integer/int__of__num h4/num/0)
% Assm: h4/rat/RAT__NMRLT0__CONG: !f1. h4/integer/int__lt (h4/frac/frac__nmr (h4/rat/rep__rat (h4/rat/abs__rat f1))) (h4/integer/int__of__num h4/num/0) <=> h4/integer/int__lt (h4/frac/frac__nmr f1) (h4/integer/int__of__num 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/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/rat/rat__equiv__def: !f2 f1. h4/rat/rat__equiv f1 f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm: h4/frac/frac__ainv__def: !f1. h4/frac/frac__ainv f1 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__neg (h4/frac/frac__nmr f1)) (h4/frac/frac__dnm f1))
% Assm: h4/numRing/num__rewrites_c75: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> T
% Assm: h4/numRing/num__rewrites_c28: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/numRing/num__rewrites_c8: !n m. h4/arithmetic/_2D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m)
% Assm: h4/integerRing/int__rewrites_c30: !n m. h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m) = h4/bool/COND (h4/prim__rec/_3C m n) (h4/arithmetic/NUMERAL (h4/numeral/iSUB T n m)) h4/num/0
% Assm: h4/integerRing/int__rewrites_c19: !n m. h4/prim__rec/_3C (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/prim__rec/_3C n m
% Assm: h4/integerRing/int__rewrites_c23: !n. h4/arithmetic/_3C_3D h4/arithmetic/ZERO n <=> T
% Assm: h4/integerRing/int__rewrites_c18: !n. h4/prim__rec/_3C n h4/arithmetic/ZERO <=> F
% Assm: h4/integerRing/int__rewrites_c14: h4/integer/int__0 = h4/integer/int__of__num h4/num/0
% Assm: h4/integerRing/int__rewrites_c26: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/integerRing/int__rewrites_c15: h4/integer/int__1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/integerRing/int__rewrites_c2: !n m. h4/integer/int__add (h4/integer/int__of__num n) (h4/integer/int__neg (h4/integer/int__of__num m)) = h4/bool/COND (h4/arithmetic/_3C_3D m n) (h4/integer/int__of__num (h4/arithmetic/_2D n m)) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/_2D m n)))
% Assm: h4/integerRing/int__ring__thms_c53: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pplus pl pr) = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm: h4/integerRing/int__ring__thms_c52: !c. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c48: !vm. h4/integerRing/int__r__interp__cs vm h4/canonical/Nil__monom = h4/integer/int__0
% Assm: h4/integerRing/int__ring__thms_c54: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pmult pl pr) = h4/integerRing/int__r__canonical__sum__prod (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm: h4/integerRing/int__ring__thms_c51: !i. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/integerRing/int__ring__thms_c36: !t l c. h4/integerRing/int__r__canonical__sum__simplify (h4/canonical/Cons__monom c l t) = h4/bool/COND (c = h4/integer/int__0) (h4/integerRing/int__r__canonical__sum__simplify t) (h4/bool/COND (c = h4/integer/int__1) (h4/canonical/Cons__varlist l (h4/integerRing/int__r__canonical__sum__simplify t)) (h4/canonical/Cons__monom c l (h4/integerRing/int__r__canonical__sum__simplify t)))
% Assm: h4/integerRing/int__ring__thms_c35: !s2. h4/integerRing/int__r__canonical__sum__prod h4/canonical/Nil__monom s2 = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c33: !t1 s2 l1 c1. h4/integerRing/int__r__canonical__sum__prod (h4/canonical/Cons__monom c1 l1 t1) s2 = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__r__canonical__sum__scalar3 c1 l1 s2) (h4/integerRing/int__r__canonical__sum__prod t1 s2)
% Assm: h4/integerRing/int__ring__thms_c55: !p. h4/integerRing/int__polynom__normalize (h4/ringNorm/Popp p) = h4/integerRing/int__r__canonical__sum__scalar3 (h4/integer/int__neg h4/integer/int__1) h4/list/NIL (h4/integerRing/int__polynom__normalize p)
% Assm: h4/integerRing/int__ring__thms_c38: h4/integerRing/int__r__canonical__sum__simplify h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c34: !t1 s2 l1. h4/integerRing/int__r__canonical__sum__prod (h4/canonical/Cons__varlist l1 t1) s2 = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__r__canonical__sum__scalar2 l1 s2) (h4/integerRing/int__r__canonical__sum__prod t1 s2)
% Assm: h4/integerRing/int__ring__thms_c32: !l0 c0. h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c28: !t l0 l. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar2 l0 t)
% Assm: h4/integerRing/int__ring__thms_c23: !l1. h4/integerRing/int__r__varlist__insert l1 h4/canonical/Nil__monom = h4/canonical/Cons__varlist l1 h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c15: !s1. h4/integerRing/int__r__canonical__sum__merge s1 h4/canonical/Nil__monom = s1
% Assm: h4/integerRing/int__ring__thms_c29: !l0. h4/integerRing/int__r__canonical__sum__scalar2 l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c20: !l1 c1. h4/integerRing/int__r__monom__insert c1 l1 h4/canonical/Nil__monom = h4/canonical/Cons__monom c1 l1 h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c13: !t2 t1 l2 l1 c2. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add h4/integer/int__1 c2) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% Goal: !n. h4/rat/rat__of__num (h4/num/SUC n) = h4/rat/rat__add (h4/rat/rat__of__num n) h4/rat/rat__1
%   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_rats_ratu_u_ofu_u_numu_u_defu_c2]: !n. happ h4/rat/rat__of__num (h4/num/SUC (h4/num/SUC n)) = h4/rat/rat__add (happ h4/rat/rat__of__num (h4/num/SUC n)) h4/rat/rat__1
% Assm [h4s_rats_RATu_u_OFu_u_NUMu_u_CALCULATE]: !n1. happ h4/rat/rat__of__num n1 = happ h4/rat/abs__rat (h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num n1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_primitiveu_u_def]: !_4. (!rat__of__num n. happ (happ _4 rat__of__num) n = h4/combin/I (h4/rat/rat__add (happ rat__of__num (h4/num/SUC n)) h4/rat/rat__1)) ==> (!_3. (!rat__of__num v. happ (happ _3 rat__of__num) v = h4/arithmetic/num__CASE v (h4/combin/I h4/rat/rat__1) (happ _4 rat__of__num)) ==> (!_2. (!rat__of__num a. happ (happ _2 rat__of__num) a = h4/arithmetic/num__CASE a (h4/combin/I h4/rat/rat__0) (happ _3 rat__of__num)) ==> (!_1. (!rat__of__num. happ _1 rat__of__num = happ _2 rat__of__num) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!n. happ (happ R (h4/num/SUC n)) (h4/num/SUC (h4/num/SUC n)))) ==> h4/rat/rat__of__num = h4/relation/WFREC (h4/min/_40 _0) _1))))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c3]: !n. happ h4/rat/rat__of__num (h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n))) = h4/rat/rat__add (happ h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n))) h4/rat/rat__1
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c2]: !n. happ h4/rat/rat__of__num (h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) = h4/rat/rat__add (happ h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) h4/rat/rat__1
% Assm [h4s_rats_ratu_u_1u_u_def]: h4/rat/rat__1 = happ h4/rat/abs__rat h4/frac/frac__1
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_defu_c1]: happ h4/rat/rat__of__num (h4/num/SUC h4/num/0) = h4/rat/rat__1
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c1]: happ h4/rat/rat__of__num (h4/num/SUC h4/num/0) = h4/rat/rat__1
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% 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_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c0]: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c1]: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = happ f n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C (h4/num/SUC m)) (h4/num/SUC n) <=> happ (happ h4/prim__rec/_3C m) n
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS]: !g f. (!n. happ g (h4/num/SUC n) = happ (happ f n) (h4/num/SUC n)) <=> (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = happ (happ f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = happ (happ f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm [h4s_rats_RATu_u_ADDu_u_RINV]: !a. h4/rat/rat__add a (h4/rat/rat__ainv a) = happ h4/rat/rat__of__num h4/num/0
% Assm [h4s_rats_RATu_u_ADDu_u_RID]: !a. h4/rat/rat__add a (happ h4/rat/rat__of__num h4/num/0) = a
% Assm [h4s_rats_RATu_u_ADDu_u_COMM]: !b a. h4/rat/rat__add a b = h4/rat/rat__add b a
% Assm [h4s_rats_RATu_u_ADDu_u_LINV]: !a. h4/rat/rat__add (h4/rat/rat__ainv a) a = happ h4/rat/rat__of__num h4/num/0
% Assm [h4s_fracs_NMR]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__nmr (h4/frac/abs__frac (h4/pair/_2C a b)) = a
% Assm [h4s_rats_RATu_u_ADDu_u_LID]: !a. h4/rat/rat__add (happ h4/rat/rat__of__num h4/num/0) a = a
% Assm [h4s_rats_RATu_u_LES0u_u_LES0u_u_ADD]: !r2 r1. h4/rat/rat__les r1 (happ h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__les r2 (happ h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__les (h4/rat/rat__add r1 r2) (happ h4/rat/rat__of__num h4/num/0)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_rats_RATu_u_ADDu_u_CALCULATE]: !f2 f1. h4/rat/rat__add (happ h4/rat/abs__rat f1) (happ h4/rat/abs__rat f2) = happ h4/rat/abs__rat (h4/frac/frac__add f1 f2)
% Assm [h4s_integers_INTu_u_LTu_u_REDUCEu_c0]: !n. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) <=> T
% Assm [h4s_fracs_DNM]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__dnm (h4/frac/abs__frac (h4/pair/_2C a b)) = b
% Assm [h4s_fracs_fracu_u_addu_u_def]: !f2 f1. h4/frac/frac__add f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__add (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2)) (h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1))) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% Assm [h4s_fracs_fracu_u_0u_u_def]: h4/frac/frac__0 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_fracs_fracu_u_1u_u_def]: h4/frac/frac__1 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_integers_INT]: !n. h4/integer/int__of__num (h4/num/SUC n) = h4/integer/int__add (h4/integer/int__of__num n) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_integers_INTu_u_MUL]: !n m. h4/integer/int__mul (h4/integer/int__of__num m) (h4/integer/int__of__num n) = h4/integer/int__of__num (h4/arithmetic/_2A m n)
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_ind]: !P. happ P h4/num/0 /\ happ P (h4/num/SUC h4/num/0) /\ (!n. happ P (h4/num/SUC n) ==> happ P (h4/num/SUC (h4/num/SUC n))) ==> (!v. happ P v)
% Assm [h4s_rats_ratu_u_0u_u_def]: h4/rat/rat__0 = happ h4/rat/abs__rat h4/frac/frac__0
% Assm [h4s_arithmetics_MULTu_u_RIGHTu_u_1]: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_intu_u_ariths_INTu_u_NUMu_u_FORALL]: !P. (!n. happ P (h4/integer/int__of__num n)) <=> (!x. h4/integer/int__le (h4/integer/int__of__num h4/num/0) x ==> happ P x)
% Assm [h4s_integers_INTu_u_MULu_u_LID]: !x. h4/integer/int__mul (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm [h4s_numerals_numeralu_u_distribu_c13]: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% 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_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_rats_ratu_u_00]: happ h4/rat/rat__of__num h4/num/0 = happ h4/rat/abs__rat h4/frac/frac__0
% Assm [h4s_rats_ratu_u_addu_u_def]: !r2 r1. h4/rat/rat__add r1 r2 = happ h4/rat/abs__rat (h4/frac/frac__add (happ h4/rat/rep__rat r1) (happ h4/rat/rep__rat r2))
% Assm [h4s_rats_RATu_u_ADDu_u_CONGu_c1]: !y x. happ h4/rat/abs__rat (h4/frac/frac__add x (happ h4/rat/rep__rat (happ h4/rat/abs__rat y))) = happ h4/rat/abs__rat (h4/frac/frac__add x y)
% Assm [h4s_fracs_FRACu_u_ADDu_u_RID]: !a. h4/frac/frac__add a h4/frac/frac__0 = a
% Assm [h4s_rats_ratu_u_QUOTIENT]: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm [h4s_quotients_QUOTIENTu_u_def]: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. happ abs (happ rep a) = a) /\ (!a. happ (happ R (happ rep a)) (happ rep a)) /\ (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_fracs_FRACu_u_DNMPOS]: !f. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/frac/frac__dnm f)
% Assm [h4s_intExtensions_INTu_u_MULu_u_POSu_u_SIGN]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) a ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul a b)
% Assm [h4s_fracs_FRACu_u_ADDu_u_CALCULATE]: !b2 b1 a2 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b2 ==> h4/frac/frac__add (h4/frac/abs__frac (h4/pair/_2C a1 b1)) (h4/frac/abs__frac (h4/pair/_2C a2 b2)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__add (h4/integer/int__mul a1 b2) (h4/integer/int__mul a2 b1)) (h4/integer/int__mul b1 b2))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_fracs_FRAC]: !f. h4/frac/abs__frac (h4/pair/_2C (h4/frac/frac__nmr f) (h4/frac/frac__dnm f)) = f
% Assm [h4s_intExtensions_INTu_u_LTu_u_ADDu_u_NEG]: !y x. h4/integer/int__lt x (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__lt y (h4/integer/int__of__num h4/num/0) ==> h4/integer/int__lt (h4/integer/int__add x y) (h4/integer/int__of__num h4/num/0)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_rats_RATu_u_LES0u_u_NMR]: !r1. h4/rat/rat__les r1 (happ h4/rat/rat__of__num h4/num/0) <=> h4/integer/int__lt (h4/rat/rat__nmr r1) (h4/integer/int__of__num h4/num/0)
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_rats_ratu_u_nmru_u_def]: !r. h4/rat/rat__nmr r = h4/frac/frac__nmr (happ h4/rat/rep__rat r)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_rats_RAT]: !r. happ h4/rat/abs__rat (happ h4/rat/rep__rat r) = r
% Assm [h4s_integers_INTu_u_MULu_u_SIGNu_u_CASESu_c1]: !q p. h4/integer/int__lt (h4/integer/int__mul p q) (h4/integer/int__of__num h4/num/0) <=> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) p /\ h4/integer/int__lt q (h4/integer/int__of__num h4/num/0) \/ h4/integer/int__lt p (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__lt (h4/integer/int__of__num h4/num/0) q
% Assm [h4s_integers_INTu_u_MULu_u_SIGNu_u_CASESu_c0]: !q p. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul p q) <=> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) p /\ h4/integer/int__lt (h4/integer/int__of__num h4/num/0) q \/ h4/integer/int__lt p (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__lt q (h4/integer/int__of__num h4/num/0)
% Assm [h4s_rats_RATu_u_NMRLT0u_u_CONG]: !f1. h4/integer/int__lt (h4/frac/frac__nmr (happ h4/rat/rep__rat (happ h4/rat/abs__rat f1))) (h4/integer/int__of__num h4/num/0) <=> h4/integer/int__lt (h4/frac/frac__nmr f1) (h4/integer/int__of__num 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_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_rats_ratu_u_equivu_u_def]: !f2 f1. happ (happ h4/rat/rat__equiv f1) f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm [h4s_fracs_fracu_u_ainvu_u_def]: !f1. h4/frac/frac__ainv f1 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__neg (h4/frac/frac__nmr f1)) (h4/frac/frac__dnm f1))
% Assm [h4s_numRings_numu_u_rewritesu_c75]: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> T
% Assm [h4s_numRings_numu_u_rewritesu_c28]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_numRings_numu_u_rewritesu_c8]: !n m. h4/arithmetic/_2D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m)
% Assm [h4s_integerRings_intu_u_rewritesu_c30]: !n m. h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m) = h4/bool/COND (happ (happ h4/prim__rec/_3C m) n) (h4/arithmetic/NUMERAL (h4/numeral/iSUB T n m)) h4/num/0
% Assm [h4s_integerRings_intu_u_rewritesu_c19]: !n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/BIT1 n)) (h4/arithmetic/BIT1 m) <=> happ (happ h4/prim__rec/_3C n) m
% Assm [h4s_integerRings_intu_u_rewritesu_c23]: !n. h4/arithmetic/_3C_3D h4/arithmetic/ZERO n <=> T
% Assm [h4s_integerRings_intu_u_rewritesu_c18]: !n. happ (happ h4/prim__rec/_3C n) h4/arithmetic/ZERO <=> F
% Assm [h4s_integerRings_intu_u_rewritesu_c14]: h4/integer/int__0 = h4/integer/int__of__num h4/num/0
% Assm [h4s_integerRings_intu_u_rewritesu_c26]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_integerRings_intu_u_rewritesu_c15]: h4/integer/int__1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_integerRings_intu_u_rewritesu_c2]: !n m. h4/integer/int__add (h4/integer/int__of__num n) (h4/integer/int__neg (h4/integer/int__of__num m)) = h4/bool/COND (h4/arithmetic/_3C_3D m n) (h4/integer/int__of__num (h4/arithmetic/_2D n m)) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/_2D m n)))
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c53]: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pplus pl pr) = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c52]: !c. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c48]: !vm. h4/integerRing/int__r__interp__cs vm h4/canonical/Nil__monom = h4/integer/int__0
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c54]: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pmult pl pr) = h4/integerRing/int__r__canonical__sum__prod (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c51]: !i. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c36]: !t l c. ?v. (v <=> c = h4/integer/int__1) /\ (?v'. (v' <=> c = h4/integer/int__0) /\ h4/integerRing/int__r__canonical__sum__simplify (h4/canonical/Cons__monom c l t) = h4/bool/COND v' (h4/integerRing/int__r__canonical__sum__simplify t) (h4/bool/COND v (h4/canonical/Cons__varlist l (h4/integerRing/int__r__canonical__sum__simplify t)) (h4/canonical/Cons__monom c l (h4/integerRing/int__r__canonical__sum__simplify t))))
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c35]: !s2. h4/integerRing/int__r__canonical__sum__prod h4/canonical/Nil__monom s2 = h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c33]: !t1 s2 l1 c1. h4/integerRing/int__r__canonical__sum__prod (h4/canonical/Cons__monom c1 l1 t1) s2 = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__r__canonical__sum__scalar3 c1 l1 s2) (h4/integerRing/int__r__canonical__sum__prod t1 s2)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c55]: !p. h4/integerRing/int__polynom__normalize (h4/ringNorm/Popp p) = h4/integerRing/int__r__canonical__sum__scalar3 (h4/integer/int__neg h4/integer/int__1) h4/list/NIL (h4/integerRing/int__polynom__normalize p)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c38]: h4/integerRing/int__r__canonical__sum__simplify h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c34]: !t1 s2 l1. h4/integerRing/int__r__canonical__sum__prod (h4/canonical/Cons__varlist l1 t1) s2 = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__r__canonical__sum__scalar2 l1 s2) (h4/integerRing/int__r__canonical__sum__prod t1 s2)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c32]: !l0 c0. h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c28]: !t l0 l. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar2 l0 t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c23]: !l1. h4/integerRing/int__r__varlist__insert l1 h4/canonical/Nil__monom = h4/canonical/Cons__varlist l1 h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c15]: !s1. h4/integerRing/int__r__canonical__sum__merge s1 h4/canonical/Nil__monom = s1
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c29]: !l0. h4/integerRing/int__r__canonical__sum__scalar2 l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c20]: !l1 c1. h4/integerRing/int__r__monom__insert c1 l1 h4/canonical/Nil__monom = h4/canonical/Cons__monom c1 l1 h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c13]: !t2 t1 l2 l1 c2. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add h4/integer/int__1 c2) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% Goal: !n. happ h4/rat/rat__of__num (h4/num/SUC n) = h4/rat/rat__add (happ h4/rat/rat__of__num n) h4/rat/rat__1
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_Q1172841,TV_Q1172837]: ![V_f, V_g]: (![V_x]: s(TV_Q1172837,happ(s(t_fun(TV_Q1172841,TV_Q1172837),V_f),s(TV_Q1172841,V_x))) = s(TV_Q1172837,happ(s(t_fun(TV_Q1172841,TV_Q1172837),V_g),s(TV_Q1172841,V_x))) => s(t_fun(TV_Q1172841,TV_Q1172837),V_f) = s(t_fun(TV_Q1172841,TV_Q1172837),V_g))).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_c2, axiom, ![V_n]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_1)))).
fof(ah4s_rats_RATu_u_OFu_u_NUMu_u_CALCULATE, axiom, ![V_n1]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,V_n1))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n1))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_rats_ratu_u_ofu_u_numu_u_primitiveu_u_def, axiom, ![V_uu_4]: (![V_ratu_u_ofu_u_num, V_n]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_4),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))),s(t_h4s_nums_num,V_n))) = s(t_h4s_rats_rat,h4s_combins_i(s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_1))))) => ![V_uu_3]: (![V_ratu_u_ofu_u_num, V_v]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_3),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))),s(t_h4s_nums_num,V_v))) = s(t_h4s_rats_rat,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,V_v),s(t_h4s_rats_rat,h4s_combins_i(s(t_h4s_rats_rat,h4s_rats_ratu_u_1))),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_4),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))))) => ![V_uu_2]: (![V_ratu_u_ofu_u_num, V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_2),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))),s(t_h4s_nums_num,V_a))) = s(t_h4s_rats_rat,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,V_a),s(t_h4s_rats_rat,h4s_combins_i(s(t_h4s_rats_rat,h4s_rats_ratu_u_0))),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_3),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))))) => ![V_uu_1]: (![V_ratu_u_ofu_u_num]: s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_1),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))) = s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_2),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R)))) & ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R),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,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))))) => s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num) = s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_relations_wfrec(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_1))))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c3, axiom, ![V_n]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_1)))).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c2, axiom, ![V_n]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),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,V_n))))))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_1)))).
fof(ah4s_rats_ratu_u_1u_u_def, axiom, s(t_h4s_rats_rat,h4s_rats_ratu_u_1) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_1)))).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_c1, axiom, s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_1)).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c1, axiom, s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_1)).
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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(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_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_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_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: 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_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),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),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(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_n)))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(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_y))),s(TV_u_27a,V_z)))) => 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_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(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_SYM, axiom, ![V_n, V_m]: s(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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_arithmetics_numu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,V_v)).
fof(ah4s_arithmetics_numu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),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_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
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_arithmetics_SUCu_u_ELIMu_u_NUMERALS, axiom, ![TV_u_27a]: ![V_g, V_f]: (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) <=> (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(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))))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_rats_RATu_u_ADDu_u_RINV, axiom, ![V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_a))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_rats_RATu_u_ADDu_u_RID, axiom, ![V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_rats_rat,V_a)).
fof(ah4s_rats_RATu_u_ADDu_u_COMM, axiom, ![V_b, V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,V_b))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_b),s(t_h4s_rats_rat,V_a)))).
fof(ah4s_rats_RATu_u_ADDu_u_LINV, axiom, ![V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,V_a))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_fracs_NMR, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))) = s(t_h4s_integers_int,V_a))).
fof(ah4s_rats_RATu_u_ADDu_u_LID, axiom, ![V_a]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,V_a))) = s(t_h4s_rats_rat,V_a)).
fof(ah4s_rats_RATu_u_LES0u_u_LES0u_u_ADD, axiom, ![V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0)))))) => (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0)))))) => p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0))))))))).
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_rats_RATu_u_ADDu_u_CALCULATE, axiom, ![V_f2, V_f1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,V_f1))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,V_f2))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))))).
fof(ah4s_integers_INTu_u_LTu_u_REDUCEu_c0, axiom, ![V_n]: s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))))) = s(t_bool,t)).
fof(ah4s_fracs_DNM, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))) = s(t_h4s_integers_int,V_b))).
fof(ah4s_fracs_fracu_u_addu_u_def, axiom, ![V_f2, V_f1]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2))) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2)))))))))).
fof(ah4s_fracs_fracu_u_0u_u_def, axiom, s(t_h4s_fracs_frac,h4s_fracs_fracu_u_0) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_fracs_fracu_u_1u_u_def, axiom, s(t_h4s_fracs_frac,h4s_fracs_fracu_u_1) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_integers_int,h4s_integers_intu_u_ofu_u_num(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_integers_INT, axiom, ![V_n]: s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_integers_INTu_u_MUL, axiom, ![V_n, V_m]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_ind, 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)))) & (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,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,h4s_nums_suc(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,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))))))) => ![V_v]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_v)))))).
fof(ah4s_rats_ratu_u_0u_u_def, axiom, s(t_h4s_rats_rat,h4s_rats_ratu_u_0) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_0)))).
fof(ah4s_arithmetics_MULTu_u_RIGHTu_u_1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),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)).
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_intu_u_ariths_INTu_u_NUMu_u_FORALL, axiom, ![V_P]: (![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),V_P),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n)))))) <=> ![V_x]: (p(s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_x)))) => p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),V_P),s(t_h4s_integers_int,V_x))))))).
fof(ah4s_integers_INTu_u_MULu_u_LID, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_integers_int,V_x))) = s(t_h4s_integers_int,V_x)).
fof(ah4s_numerals_numeralu_u_distribu_c13, axiom, s(t_h4s_nums_num,h4s_nums_suc(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_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_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_rats_ratu_u_00, axiom, s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_0)))).
fof(ah4s_rats_ratu_u_addu_u_def, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r1))),s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r2)))))))).
fof(ah4s_rats_RATu_u_ADDu_u_CONGu_c1, axiom, ![V_y, V_x]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_x),s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,V_y))))))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_x),s(t_h4s_fracs_frac,V_y)))))).
fof(ah4s_fracs_FRACu_u_ADDu_u_RID, axiom, ![V_a]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_a),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_0))) = s(t_h4s_fracs_frac,V_a)).
fof(ah4s_rats_ratu_u_QUOTIENT, axiom, p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat))))).
fof(ah4s_quotients_QUOTIENTu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & (![V_a]: p(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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))) & ![V_r, V_s]: (p(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_r))),s(TV_u_27a,V_s)))) <=> (p(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_r))),s(TV_u_27a,V_r)))) & (p(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_s))),s(TV_u_27a,V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))))).
fof(ah4s_fracs_FRACu_u_DNMPOS, axiom, ![V_f]: p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f))))))).
fof(ah4s_intExtensions_INTu_u_MULu_u_POSu_u_SIGN, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_a)))) => (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))))).
fof(ah4s_fracs_FRACu_u_ADDu_u_CALCULATE, axiom, ![V_b2, V_b1, V_a2, V_a1]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b1)))) => (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b2)))) => s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a1),s(t_h4s_integers_int,V_b1))))),s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a2),s(t_h4s_integers_int,V_b2))))))) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a1),s(t_h4s_integers_int,V_b2))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a2),s(t_h4s_integers_int,V_b1))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_b1),s(t_h4s_integers_int,V_b2)))))))))).
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_fracs_FRAC, axiom, ![V_f]: s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f))))))) = s(t_h4s_fracs_frac,V_f)).
fof(ah4s_intExtensions_INTu_u_LTu_u_ADDu_u_NEG, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))))).
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_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_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_CONJu_u_ASSOC, 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_rats_RATu_u_LES0u_u_NMR, axiom, ![V_r1]: s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0))))) = s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_rats_ratu_u_nmr(s(t_h4s_rats_rat,V_r1))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_rats_ratu_u_nmru_u_def, axiom, ![V_r]: s(t_h4s_integers_int,h4s_rats_ratu_u_nmr(s(t_h4s_rats_rat,V_r))) = s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r)))))).
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_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_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_rats_RAT, axiom, ![V_r]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r))))) = s(t_h4s_rats_rat,V_r)).
fof(ah4s_integers_INTu_u_MULu_u_SIGNu_u_CASESu_c1, axiom, ![V_q, V_p]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,V_q))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) <=> ((p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_p)))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_q),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))) | (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_q)))))))).
fof(ah4s_integers_INTu_u_MULu_u_SIGNu_u_CASESu_c0, axiom, ![V_q, V_p]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,V_q)))))) <=> ((p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_p)))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_q))))) | (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_q),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_rats_RATu_u_NMRLT0u_u_CONG, axiom, ![V_f1]: s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,V_f1))))))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_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_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_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_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_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_sats_NOTu_u_NOT, axiom, ![V_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_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_rats_ratu_u_equivu_u_def, axiom, ![V_f2, V_f1]: (p(s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_f1))),s(t_h4s_fracs_frac,V_f2)))) <=> s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))))))).
fof(ah4s_fracs_fracu_u_ainvu_u_def, axiom, ![V_f1]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_ainv(s(t_h4s_fracs_frac,V_f1))) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1)))))))).
fof(ah4s_numRings_numu_u_rewritesu_c75, axiom, ![TV_u_27a]: ![V_ALTu_u_ZERO]: (s(TV_u_27a,V_ALTu_u_ZERO) = s(TV_u_27a,V_ALTu_u_ZERO) <=> p(s(t_bool,t)))).
fof(ah4s_numRings_numu_u_rewritesu_c28, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_numRings_numu_u_rewritesu_c8, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_integerRings_intu_u_rewritesu_c30, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_integerRings_intu_u_rewritesu_c19, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))).
fof(ah4s_integerRings_intu_u_rewritesu_c23, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))) = s(t_bool,t)).
fof(ah4s_integerRings_intu_u_rewritesu_c18, axiom, ![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_integerRings_intu_u_rewritesu_c14, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_0) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_integerRings_intu_u_rewritesu_c26, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_integerRings_intu_u_rewritesu_c15, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_1) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_integerRings_intu_u_rewritesu_c2, axiom, ![V_n, V_m]: s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c53, axiom, ![V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pplus(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c52, axiom, ![V_c]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pconst(s(t_h4s_integers_int,V_c))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c48, axiom, ![V_vm]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_cs(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_integers_int,h4s_integers_intu_u_0)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c54, axiom, ![V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pmult(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c51, axiom, ![V_i]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pvar(s(t_h4s_quotes_index,V_i))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_i),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)))).
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_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_integerRings_intu_u_ringu_u_thmsu_c36, axiom, ![V_t, V_l, V_c]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_integers_int,V_c) = s(t_h4s_integers_int,h4s_integers_intu_u_1)) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_integers_int,V_c) = s(t_h4s_integers_int,h4s_integers_intu_u_0)) & s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_bools_cond(s(t_bool,V_vi_),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))))))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c35, axiom, ![V_s2]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c33, axiom, ![V_t1, V_s2, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3(s(t_h4s_integers_int,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c55, axiom, ![V_p]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_popp(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_1))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c38, axiom, s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c34, axiom, ![V_t1, V_s2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c32, axiom, ![V_l0, V_c0]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3(s(t_h4s_integers_int,V_c0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c28, axiom, ![V_t, V_l0, V_l]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_varlistu_u_insert(s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c23, axiom, ![V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_varlistu_u_insert(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c15, axiom, ![V_s1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s1)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c29, axiom, ![V_l0]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c20, axiom, ![V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_monomu_u_insert(s(t_h4s_integers_int,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c13, axiom, ![V_t2, V_t1, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_h4s_integers_int,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2)))))))).
fof(ch4s_rats_RATu_u_OFu_u_NUMu_c1, conjecture, ![V_n]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,V_n))),s(t_h4s_rats_rat,h4s_rats_ratu_u_1)))).
