%   ORIGINAL: h4/rat/RAT__SAVE__NUM
% 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__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_c1: !n. h4/rat/rat__of__num (h4/num/SUC n) = h4/rat/rat__add (h4/rat/rat__of__num n) h4/rat/rat__1
% Assm: h4/rat/RAT__SAVE__MINV: !b1 a1. ~(h4/rat/abs__rat (h4/frac/frac__save a1 b1) = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__minv (h4/rat/abs__rat (h4/frac/frac__save a1 b1)) = h4/rat/abs__rat (h4/frac/frac__save (h4/integer/int__mul (h4/intExtension/SGN a1) (h4/integer/int__add (h4/integer/int__of__num b1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) (h4/integer/Num (h4/integer/int__sub (h4/integer/ABS a1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))))
% Assm: h4/rat/RAT__SAVE__TO__CONS: !b1 a1. h4/rat/abs__rat (h4/frac/frac__save a1 b1) = h4/rat/rat__cons a1 (h4/integer/int__add (h4/integer/int__of__num b1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% 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/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% 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/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% 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/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/arithmetic/num__case__def_c0: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% 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/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n 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/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/arithmetic/num__case__def_c1: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = f n
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/rat/RAT__ADD__LID: !a. h4/rat/rat__add (h4/rat/rat__of__num h4/num/0) a = a
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/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/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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/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/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/rat/rat__1__def: h4/rat/rat__1 = h4/rat/abs__rat h4/frac/frac__1
% 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/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/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/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/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/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/numeral/numeral__distrib_c13: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/arithmetic/MULT__RIGHT__1: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% 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/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/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/rat/RAT__MINV__CALCULATE: !f1. ~(h4/integer/int__of__num h4/num/0 = h4/frac/frac__nmr f1) ==> h4/rat/rat__minv (h4/rat/abs__rat f1) = h4/rat/abs__rat (h4/frac/frac__minv f1)
% Assm: h4/rat/RAT__NMREQ0__CONG: !f1. h4/frac/frac__nmr (h4/rat/rep__rat (h4/rat/abs__rat f1)) = h4/integer/int__of__num h4/num/0 <=> h4/frac/frac__nmr f1 = h4/integer/int__of__num h4/num/0
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/rat/RAT__EQ0__NMR: !r1. r1 = h4/rat/rat__of__num h4/num/0 <=> h4/rat/rat__nmr r1 = h4/integer/int__of__num h4/num/0
% Assm: h4/rat/rat__nmr__def: !r. h4/rat/rat__nmr r = h4/frac/frac__nmr (h4/rat/rep__rat r)
% Assm: h4/frac/FRAC__MINV__SAVE: !b1 a1. ~(a1 = h4/integer/int__of__num h4/num/0) ==> h4/frac/frac__minv (h4/frac/frac__save a1 b1) = h4/frac/frac__save (h4/integer/int__mul (h4/intExtension/SGN a1) (h4/integer/int__add (h4/integer/int__of__num b1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) (h4/integer/Num (h4/integer/int__sub (h4/integer/ABS a1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))
% Assm: h4/frac/FRAC__NMR__SAVE: !b1 a1. h4/frac/frac__nmr (h4/frac/frac__save a1 b1) = a1
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/sat/dc__imp: !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/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/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/rat/RAT__ABS__EQUIV: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat f2 <=> h4/rat/rat__equiv f1 f2
% Assm: h4/rat/rat__cons__def: !nmr dnm. h4/rat/rat__cons nmr dnm = h4/rat/abs__rat (h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/integer/int__mul (h4/intExtension/SGN nmr) (h4/intExtension/SGN dnm)) (h4/integer/ABS nmr)) (h4/integer/ABS dnm)))
% 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__save__def: !nmr dnm. h4/frac/frac__save nmr dnm = h4/frac/abs__frac (h4/pair/_2C nmr (h4/integer/int__add (h4/integer/int__of__num dnm) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/int__arith/le__context__rwt2: !y x c. h4/integer/int__le (h4/integer/int__of__num h4/num/0) (h4/integer/int__add c x) ==> h4/integer/int__lt y (h4/integer/int__neg x) ==> (h4/integer/int__le (h4/integer/int__of__num h4/num/0) (h4/integer/int__add (h4/integer/int__neg c) y) <=> F)
% Assm: h4/int__arith/le__move__all__right: !y x. h4/integer/int__le x y <=> h4/integer/int__le (h4/integer/int__of__num h4/num/0) (h4/integer/int__add y (h4/integer/int__neg x))
% 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/integer/INT__LT__REDUCE_c6: !n. h4/integer/int__lt (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))) (h4/integer/int__of__num h4/num/0) <=> T
% Assm: h4/integer/INT__LT__REDUCE_c5: !n. h4/integer/int__lt (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))) (h4/integer/int__of__num h4/num/0) <=> T
% Assm: h4/intExtension/SGN__def: !x. h4/intExtension/SGN x = h4/bool/COND (x = h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num h4/num/0) (h4/bool/COND (h4/integer/int__lt x (h4/integer/int__of__num h4/num/0)) (h4/integer/int__neg (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/int__arith/less__to__leq__samer: !y x. h4/integer/int__lt x y <=> h4/integer/int__le (h4/integer/int__add x (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) y
% Assm: h4/integer/INT__ADD__REDUCE_c7: !n m. h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL n))) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL m))) = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))))
% Assm: h4/integer/INT__ADD__REDUCE_c1: !p. h4/integer/int__add p (h4/integer/int__of__num h4/num/0) = p
% Assm: h4/integer/INT__INJ: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm: h4/integer/INT__NEG__0: h4/integer/int__neg (h4/integer/int__of__num h4/num/0) = h4/integer/int__of__num h4/num/0
% Assm: h4/integer/INT__ABS: !n. h4/integer/ABS n = h4/bool/COND (h4/integer/int__lt n (h4/integer/int__of__num h4/num/0)) (h4/integer/int__neg n) n
% Assm: h4/integer/INT__EQ__LMUL: !z y x. h4/integer/int__mul x y = h4/integer/int__mul x z <=> x = h4/integer/int__of__num h4/num/0 \/ y = z
% Assm: h4/integer/INT__NEG__MUL2: !y x. h4/integer/int__mul (h4/integer/int__neg x) (h4/integer/int__neg y) = h4/integer/int__mul x y
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/integer/INT__NEG__LMUL: !y x. h4/integer/int__neg (h4/integer/int__mul x y) = h4/integer/int__mul (h4/integer/int__neg x) y
% Assm: h4/integer/INT__MUL__RZERO: !x. h4/integer/int__mul x (h4/integer/int__of__num h4/num/0) = h4/integer/int__of__num h4/num/0
% Assm: h4/integer/INT__NEG__ADD: !y x. h4/integer/int__neg (h4/integer/int__add x y) = h4/integer/int__add (h4/integer/int__neg x) (h4/integer/int__neg y)
% Assm: h4/integer/INT__ADD__REDUCE_c0: !p. h4/integer/int__add (h4/integer/int__of__num h4/num/0) p = p
% Assm: h4/integer/INT__NEG__RMUL: !y x. h4/integer/int__neg (h4/integer/int__mul x y) = h4/integer/int__mul x (h4/integer/int__neg y)
% Assm: h4/integer/INT__MUL__LZERO: !x. h4/integer/int__mul (h4/integer/int__of__num h4/num/0) x = h4/integer/int__of__num h4/num/0
% Assm: h4/integer/INT__EQ__RADD: !z y x. h4/integer/int__add x z = h4/integer/int__add y z <=> x = y
% Assm: h4/integer/INT__ADD__ASSOC: !z y x. h4/integer/int__add x (h4/integer/int__add y z) = h4/integer/int__add (h4/integer/int__add x y) z
% Assm: h4/integer/INT__ADD__COMM: !y x. h4/integer/int__add x y = h4/integer/int__add y x
% Assm: h4/integer/int__le0: !y x. h4/integer/int__le x y <=> ~h4/integer/int__lt y x
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/integer/INT__ADD__RID: !x. h4/integer/int__add x (h4/integer/int__of__num h4/num/0) = x
% Assm: h4/numeral/numeral__add_c1: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/integer/INT__MUL__RID: !x. h4/integer/int__mul x (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Goal: !n. h4/rat/rat__of__num n = h4/rat/abs__rat (h4/frac/frac__save (h4/integer/int__of__num n) h4/num/0)
%   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_CALCULATE]: !n1. happ 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 [h4s_rats_RATu_u_OFu_u_NUMu_c1]: !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
% Assm [h4s_rats_RATu_u_SAVEu_u_MINV]: !b1 a1. ~(h4/rat/abs__rat (h4/frac/frac__save a1 b1) = happ h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__minv (h4/rat/abs__rat (h4/frac/frac__save a1 b1)) = h4/rat/abs__rat (h4/frac/frac__save (h4/integer/int__mul (h4/intExtension/SGN a1) (h4/integer/int__add (h4/integer/int__of__num b1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) (h4/integer/Num (h4/integer/int__sub (h4/integer/ABS a1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))))
% Assm [h4s_rats_RATu_u_SAVEu_u_TOu_u_CONS]: !b1 a1. h4/rat/abs__rat (h4/frac/frac__save a1 b1) = h4/rat/rat__cons a1 (h4/integer/int__add (h4/integer/int__of__num b1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% 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_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_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% 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_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% 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_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% 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_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R 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_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_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n 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_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_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_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_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_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_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_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 = h4/rat/abs__rat h4/frac/frac__0
% Assm [h4s_rats_ratu_u_1u_u_def]: h4/rat/rat__1 = h4/rat/abs__rat h4/frac/frac__1
% 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_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_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_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_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_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_numerals_numeralu_u_distribu_c13]: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% 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_rats_RATu_u_ADDu_u_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 [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_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_rats_RATu_u_MINVu_u_CALCULATE]: !f1. ~(h4/integer/int__of__num h4/num/0 = h4/frac/frac__nmr f1) ==> h4/rat/rat__minv (h4/rat/abs__rat f1) = h4/rat/abs__rat (h4/frac/frac__minv f1)
% Assm [h4s_rats_RATu_u_NMREQ0u_u_CONG]: !f1. h4/frac/frac__nmr (h4/rat/rep__rat (h4/rat/abs__rat f1)) = h4/integer/int__of__num h4/num/0 <=> h4/frac/frac__nmr f1 = h4/integer/int__of__num h4/num/0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_rats_RATu_u_EQ0u_u_NMR]: !r1. r1 = happ h4/rat/rat__of__num h4/num/0 <=> h4/rat/rat__nmr r1 = h4/integer/int__of__num h4/num/0
% Assm [h4s_rats_ratu_u_nmru_u_def]: !r. h4/rat/rat__nmr r = h4/frac/frac__nmr (h4/rat/rep__rat r)
% Assm [h4s_fracs_FRACu_u_MINVu_u_SAVE]: !b1 a1. ~(a1 = h4/integer/int__of__num h4/num/0) ==> h4/frac/frac__minv (h4/frac/frac__save a1 b1) = h4/frac/frac__save (h4/integer/int__mul (h4/intExtension/SGN a1) (h4/integer/int__add (h4/integer/int__of__num b1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) (h4/integer/Num (h4/integer/int__sub (h4/integer/ABS a1) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))
% Assm [h4s_fracs_FRACu_u_NMRu_u_SAVE]: !b1 a1. h4/frac/frac__nmr (h4/frac/frac__save a1 b1) = a1
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_sats_dcu_u_imp]: !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_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_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_rats_RATu_u_ABSu_u_EQUIV]: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat f2 <=> h4/rat/rat__equiv f1 f2
% Assm [h4s_rats_ratu_u_consu_u_def]: !nmr dnm. h4/rat/rat__cons nmr dnm = h4/rat/abs__rat (h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/integer/int__mul (h4/intExtension/SGN nmr) (h4/intExtension/SGN dnm)) (h4/integer/ABS nmr)) (h4/integer/ABS dnm)))
% Assm [h4s_rats_ratu_u_equivu_u_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 [h4s_fracs_fracu_u_saveu_u_def]: !nmr dnm. h4/frac/frac__save nmr dnm = h4/frac/abs__frac (h4/pair/_2C nmr (h4/integer/int__add (h4/integer/int__of__num dnm) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_intu_u_ariths_leu_u_contextu_u_rwt2]: !y x c. h4/integer/int__le (h4/integer/int__of__num h4/num/0) (h4/integer/int__add c x) ==> h4/integer/int__lt y (h4/integer/int__neg x) ==> (h4/integer/int__le (h4/integer/int__of__num h4/num/0) (h4/integer/int__add (h4/integer/int__neg c) y) <=> F)
% Assm [h4s_intu_u_ariths_leu_u_moveu_u_allu_u_right]: !y x. h4/integer/int__le x y <=> h4/integer/int__le (h4/integer/int__of__num h4/num/0) (h4/integer/int__add y (h4/integer/int__neg x))
% 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_integers_INTu_u_LTu_u_REDUCEu_c6]: !n. h4/integer/int__lt (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))) (h4/integer/int__of__num h4/num/0) <=> T
% Assm [h4s_integers_INTu_u_LTu_u_REDUCEu_c5]: !n. h4/integer/int__lt (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))) (h4/integer/int__of__num h4/num/0) <=> T
% Assm [h4s_intExtensions_SGNu_u_def]: !x. ?v. (v <=> x = h4/integer/int__of__num h4/num/0) /\ h4/intExtension/SGN x = h4/bool/COND v (h4/integer/int__of__num h4/num/0) (h4/bool/COND (h4/integer/int__lt x (h4/integer/int__of__num h4/num/0)) (h4/integer/int__neg (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_intu_u_ariths_lessu_u_tou_u_lequ_u_samer]: !y x. h4/integer/int__lt x y <=> h4/integer/int__le (h4/integer/int__add x (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) y
% Assm [h4s_integers_INTu_u_ADDu_u_REDUCEu_c7]: !n m. h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL n))) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL m))) = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))))
% Assm [h4s_integers_INTu_u_ADDu_u_REDUCEu_c1]: !p. h4/integer/int__add p (h4/integer/int__of__num h4/num/0) = p
% Assm [h4s_integers_INTu_u_INJ]: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm [h4s_integers_INTu_u_NEGu_u_0]: h4/integer/int__neg (h4/integer/int__of__num h4/num/0) = h4/integer/int__of__num h4/num/0
% Assm [h4s_integers_INTu_u_ABS]: !n. h4/integer/ABS n = h4/bool/COND (h4/integer/int__lt n (h4/integer/int__of__num h4/num/0)) (h4/integer/int__neg n) n
% Assm [h4s_integers_INTu_u_EQu_u_LMUL]: !z y x. h4/integer/int__mul x y = h4/integer/int__mul x z <=> x = h4/integer/int__of__num h4/num/0 \/ y = z
% Assm [h4s_integers_INTu_u_NEGu_u_MUL2]: !y x. h4/integer/int__mul (h4/integer/int__neg x) (h4/integer/int__neg y) = h4/integer/int__mul x y
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_integers_INTu_u_NEGu_u_LMUL]: !y x. h4/integer/int__neg (h4/integer/int__mul x y) = h4/integer/int__mul (h4/integer/int__neg x) y
% Assm [h4s_integers_INTu_u_MULu_u_RZERO]: !x. h4/integer/int__mul x (h4/integer/int__of__num h4/num/0) = h4/integer/int__of__num h4/num/0
% Assm [h4s_integers_INTu_u_NEGu_u_ADD]: !y x. h4/integer/int__neg (h4/integer/int__add x y) = h4/integer/int__add (h4/integer/int__neg x) (h4/integer/int__neg y)
% Assm [h4s_integers_INTu_u_ADDu_u_REDUCEu_c0]: !p. h4/integer/int__add (h4/integer/int__of__num h4/num/0) p = p
% Assm [h4s_integers_INTu_u_NEGu_u_RMUL]: !y x. h4/integer/int__neg (h4/integer/int__mul x y) = h4/integer/int__mul x (h4/integer/int__neg y)
% Assm [h4s_integers_INTu_u_MULu_u_LZERO]: !x. h4/integer/int__mul (h4/integer/int__of__num h4/num/0) x = h4/integer/int__of__num h4/num/0
% Assm [h4s_integers_INTu_u_EQu_u_RADD]: !z y x. h4/integer/int__add x z = h4/integer/int__add y z <=> x = y
% Assm [h4s_integers_INTu_u_ADDu_u_ASSOC]: !z y x. h4/integer/int__add x (h4/integer/int__add y z) = h4/integer/int__add (h4/integer/int__add x y) z
% Assm [h4s_integers_INTu_u_ADDu_u_COMM]: !y x. h4/integer/int__add x y = h4/integer/int__add y x
% Assm [h4s_integers_intu_u_le0]: !y x. h4/integer/int__le x y <=> ~h4/integer/int__lt y x
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_integers_INTu_u_ADDu_u_RID]: !x. h4/integer/int__add x (h4/integer/int__of__num h4/num/0) = x
% Assm [h4s_numerals_numeralu_u_addu_c1]: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_integers_INTu_u_MULu_u_RID]: !x. h4/integer/int__mul x (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Goal: !n. happ h4/rat/rat__of__num n = h4/rat/abs__rat (h4/frac/frac__save (h4/integer/int__of__num n) h4/num/0)
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_Q1172868,TV_Q1172864]: ![V_f, V_g]: (![V_x]: s(TV_Q1172864,happ(s(t_fun(TV_Q1172868,TV_Q1172864),V_f),s(TV_Q1172868,V_x))) = s(TV_Q1172864,happ(s(t_fun(TV_Q1172868,TV_Q1172864),V_g),s(TV_Q1172868,V_x))) => s(t_fun(TV_Q1172868,TV_Q1172864),V_f) = s(t_fun(TV_Q1172868,TV_Q1172864),V_g))).
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,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_c1, 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,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)))).
fof(ah4s_rats_RATu_u_SAVEu_u_MINV, axiom, ![V_b1, V_a1]: (~ (s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,V_a1),s(t_h4s_nums_num,V_b1))))) = 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,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,V_a1),s(t_h4s_nums_num,V_b1))))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,V_a1))),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_b1))),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_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_sub(s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_a1))),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_SAVEu_u_TOu_u_CONS, axiom, ![V_b1, V_a1]: s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,V_a1),s(t_h4s_nums_num,V_b1))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_cons(s(t_h4s_integers_int,V_a1),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_b1))),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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_TRUTH, axiom, p(s(t_bool,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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_arithmetics_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_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_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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_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_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_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_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_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_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,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_0)))).
fof(ah4s_rats_ratu_u_1u_u_def, axiom, s(t_h4s_rats_rat,h4s_rats_ratu_u_1) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_1)))).
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_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_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_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_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_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_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_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_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,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f2))))) = s(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_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_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
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_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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_rats_RATu_u_MINVu_u_CALCULATE, axiom, ![V_f1]: (~ (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_nmr(s(t_h4s_fracs_frac,V_f1)))) => s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f1))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_minv(s(t_h4s_fracs_frac,V_f1))))))).
fof(ah4s_rats_RATu_u_NMREQ0u_u_CONG, axiom, ![V_f1]: (s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(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_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_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_rats_RATu_u_EQ0u_u_NMR, axiom, ![V_r1]: (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_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_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,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,V_r)))))).
fof(ah4s_fracs_FRACu_u_MINVu_u_SAVE, axiom, ![V_b1, V_a1]: (~ (s(t_h4s_integers_int,V_a1) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_fracs_frac,h4s_fracs_fracu_u_minv(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,V_a1),s(t_h4s_nums_num,V_b1))))) = s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,V_a1))),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_b1))),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_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_sub(s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_a1))),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_NMRu_u_SAVE, axiom, ![V_b1, V_a1]: s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,V_a1),s(t_h4s_nums_num,V_b1))))) = s(t_h4s_integers_int,V_a1)).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
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_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_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_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_rats_RATu_u_ABSu_u_EQUIV, axiom, ![V_f2, V_f1]: (s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f1))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f2))) <=> p(s(t_bool,h4s_rats_ratu_u_equiv(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))))).
fof(ah4s_rats_ratu_u_consu_u_def, axiom, ![V_nmr, V_dnm]: s(t_h4s_rats_rat,h4s_rats_ratu_u_cons(s(t_h4s_integers_int,V_nmr),s(t_h4s_integers_int,V_dnm))) = s(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_mul(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,V_nmr))),s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,V_dnm))))),s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_nmr))))),s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_dnm)))))))))).
fof(ah4s_rats_ratu_u_equivu_u_def, axiom, ![V_f2, V_f1]: (p(s(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_saveu_u_def, axiom, ![V_nmr, V_dnm]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,V_nmr),s(t_h4s_nums_num,V_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_nmr),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_dnm))),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_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_intu_u_ariths_leu_u_contextu_u_rwt2, axiom, ![V_y, V_x, V_c]: (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,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_c),s(t_h4s_integers_int,V_x)))))) => (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_neg(s(t_h4s_integers_int,V_x)))))) => 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,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_c))),s(t_h4s_integers_int,V_y))))) = s(t_bool,f)))).
fof(ah4s_intu_u_ariths_leu_u_moveu_u_allu_u_right, axiom, ![V_y, V_x]: s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = 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,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x)))))))).
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_integers_INTu_u_LTu_u_REDUCEu_c6, axiom, ![V_n]: s(t_bool,h4s_integers_intu_u_lt(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_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(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_nums_0))))) = s(t_bool,t)).
fof(ah4s_integers_INTu_u_LTu_u_REDUCEu_c5, axiom, ![V_n]: s(t_bool,h4s_integers_intu_u_lt(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_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(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_nums_0))))) = s(t_bool,t)).
fof(ah4s_intExtensions_SGNu_u_def, axiom, ![V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> 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)))) & s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,V_v),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_bools_cond(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))))),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_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_intu_u_ariths_lessu_u_tou_u_lequ_u_samer, axiom, ![V_y, V_x]: s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,h4s_integers_intu_u_add(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_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_integers_int,V_y)))).
fof(ah4s_integers_INTu_u_ADDu_u_REDUCEu_c7, axiom, ![V_n, V_m]: s(t_h4s_integers_int,h4s_integers_intu_u_add(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_numeral(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,h4s_arithmetics_numeral(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_numeral(s(t_h4s_nums_num,h4s_numerals_iz(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_integers_INTu_u_ADDu_u_REDUCEu_c1, axiom, ![V_p]: s(t_h4s_integers_int,h4s_integers_intu_u_add(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))))) = s(t_h4s_integers_int,V_p)).
fof(ah4s_integers_INTu_u_INJ, axiom, ![V_n, V_m]: (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_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_integers_INTu_u_NEGu_u_0, axiom, 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_nums_0))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_integers_INTu_u_ABS, axiom, ![V_n]: s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_n))) = s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_n),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_neg(s(t_h4s_integers_int,V_n))),s(t_h4s_integers_int,V_n)))).
fof(ah4s_integers_INTu_u_EQu_u_LMUL, axiom, ![V_z, V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_z))) <=> (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))) | s(t_h4s_integers_int,V_y) = s(t_h4s_integers_int,V_z)))).
fof(ah4s_integers_INTu_u_NEGu_u_MUL2, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y)))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![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))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_integers_INTu_u_NEGu_u_LMUL, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,V_y)))).
fof(ah4s_integers_INTu_u_MULu_u_RZERO, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(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))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_integers_INTu_u_NEGu_u_ADD, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_neg(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_add(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y)))))).
fof(ah4s_integers_INTu_u_ADDu_u_REDUCEu_c0, axiom, ![V_p]: 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,h4s_nums_0))),s(t_h4s_integers_int,V_p))) = s(t_h4s_integers_int,V_p)).
fof(ah4s_integers_INTu_u_NEGu_u_RMUL, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y)))))).
fof(ah4s_integers_INTu_u_MULu_u_LZERO, 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_nums_0))),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)))).
fof(ah4s_integers_INTu_u_EQu_u_RADD, axiom, ![V_z, V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_z))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))) <=> s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,V_y))).
fof(ah4s_integers_INTu_u_ADDu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(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,V_z)))).
fof(ah4s_integers_INTu_u_ADDu_u_COMM, axiom, ![V_y, V_x]: 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_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x)))).
fof(ah4s_integers_intu_u_le0, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y)))) <=> ~ (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x))))))).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(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_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(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_integers_INTu_u_ADDu_u_RID, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_add(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))))) = s(t_h4s_integers_int,V_x)).
fof(ah4s_numerals_numeralu_u_addu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
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_bools_boolu_u_caseu_u_thmu_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_integers_INTu_u_MULu_u_RID, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(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_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)).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, 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_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ch4s_rats_RATu_u_SAVEu_u_NUM, 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,V_n))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_save(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0)))))).
