%   ORIGINAL: h4/rat/RAT__AINV__MINV
% 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/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/integer/INT__ADD__RINV: !x. h4/integer/int__add x (h4/integer/int__neg x) = h4/integer/int__of__num h4/num/0
% 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/integer/INT__LNEG__UNIQ: !y x. h4/integer/int__add x y = h4/integer/int__of__num h4/num/0 <=> x = h4/integer/int__neg y
% 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/integer/INT__NEG__LT0: !x. h4/integer/int__lt (h4/integer/int__neg x) (h4/integer/int__of__num h4/num/0) <=> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) x
% 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__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__EQ__NEG: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% 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__MUL__CALCULATE_c1: !y x. h4/integer/int__mul (h4/integer/int__neg x) y = h4/integer/int__neg (h4/integer/int__mul x y)
% Assm: h4/prelim/list__merge__def_c1: !v5 v4 a__lt. h4/prelim/list__merge a__lt h4/list/NIL (h4/list/CONS v4 v5) = h4/list/CONS v4 v5
% Assm: h4/prelim/list__merge__def_c2: !y x l2 l1 a__lt. h4/prelim/list__merge a__lt (h4/list/CONS x l1) (h4/list/CONS y l2) = h4/bool/COND (a__lt x y) (h4/list/CONS x (h4/prelim/list__merge a__lt l1 (h4/list/CONS y l2))) (h4/list/CONS y (h4/prelim/list__merge a__lt (h4/list/CONS x l1) l2))
% Assm: h4/quote/index__compare__def_c1: !v10. h4/quote/index__compare h4/quote/End__idx (h4/quote/Left__idx v10) = h4/prelim/LESS
% Assm: h4/quote/index__lt__def: !i2 i1. h4/quote/index__lt i1 i2 <=> h4/quote/index__compare i1 i2 = h4/prelim/LESS
% Assm: h4/integerRing/int__ring__thms_c1: !vm p. h4/integerRing/int__interp__p vm p = h4/integerRing/int__r__interp__cs vm (h4/integerRing/int__polynom__simplify p)
% Assm: h4/integerRing/int__ring__thms_c2: !vm c. h4/integerRing/int__interp__p vm (h4/ringNorm/Pconst c) = c
% Assm: h4/integerRing/int__ring__thms_c3: !vm i. h4/integerRing/int__interp__p vm (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm: h4/integerRing/int__ring__thms_c5: !vm p2 p1. h4/integerRing/int__interp__p vm (h4/ringNorm/Pmult p1 p2) = h4/integer/int__mul (h4/integerRing/int__interp__p vm p1) (h4/integerRing/int__interp__p vm p2)
% Assm: h4/integerRing/int__ring__thms_c6: !vm p1. h4/integerRing/int__interp__p vm (h4/ringNorm/Popp p1) = h4/integer/int__neg (h4/integerRing/int__interp__p vm p1)
% Assm: h4/integerRing/int__ring__thms_c7: !x v2 v1. h4/quote/varmap__find h4/quote/End__idx (h4/quote/Node__vm x v1 v2) = x
% Assm: h4/integerRing/int__ring__thms_c9: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v1
% 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_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_c27: !t l0 l c. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__monom__insert c (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_c29: !l0. h4/integerRing/int__r__canonical__sum__scalar2 l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c30: !t l0 l c0 c. h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__monom__insert (h4/integer/int__mul c0 c) (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 t)
% Assm: h4/integerRing/int__ring__thms_c31: !t l0 l c0. h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__monom__insert c0 (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 t)
% 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_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_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_c35: !s2. h4/integerRing/int__r__canonical__sum__prod h4/canonical/Nil__monom s2 = h4/canonical/Nil__monom
% 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_c38: h4/integerRing/int__r__canonical__sum__simplify h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c39: !x vm. h4/integerRing/int__r__ivl__aux vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/integerRing/int__ring__thms_c40: !x_27 x vm t_27. h4/integerRing/int__r__ivl__aux vm x (h4/list/CONS x_27 t_27) = h4/integer/int__mul (h4/quote/varmap__find x vm) (h4/integerRing/int__r__ivl__aux vm x_27 t_27)
% Assm: h4/integerRing/int__ring__thms_c42: !x vm t. h4/integerRing/int__r__interp__vl vm (h4/list/CONS x t) = h4/integerRing/int__r__ivl__aux vm x t
% Assm: h4/integerRing/int__ring__thms_c44: !x vm t c. h4/integerRing/int__r__interp__m vm c (h4/list/CONS x t) = h4/integer/int__mul c (h4/integerRing/int__r__ivl__aux vm x t)
% Assm: h4/integerRing/int__ring__thms_c45: !vm a. h4/integerRing/int__r__ics__aux vm a h4/canonical/Nil__monom = a
% Assm: h4/integerRing/int__ring__thms_c49: !vm t l. h4/integerRing/int__r__interp__cs vm (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__vl vm l) t
% Assm: h4/integerRing/int__ring__thms_c50: !vm t l c. h4/integerRing/int__r__interp__cs vm (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__m vm c l) t
% 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/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_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_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_c56: !x. h4/integerRing/int__polynom__simplify x = h4/integerRing/int__r__canonical__sum__simplify (h4/integerRing/int__polynom__normalize x)
% Assm: h4/integerRing/int__rewrites_c7: !n m. h4/integer/int__mul (h4/integer/int__neg (h4/integer/int__of__num n)) (h4/integer/int__neg (h4/integer/int__of__num m)) = h4/integer/int__of__num (h4/arithmetic/_2A n m)
% Assm: h4/integerRing/int__rewrites_c8: !n m. h4/integer/int__of__num n = h4/integer/int__of__num m <=> n = m
% Assm: h4/integerRing/int__rewrites_c10: !n m. h4/integer/int__neg (h4/integer/int__of__num n) = h4/integer/int__of__num m <=> n = h4/num/0 /\ m = h4/num/0
% Assm: h4/integerRing/int__rewrites_c14: h4/integer/int__0 = h4/integer/int__of__num h4/num/0
% 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/numRing/num__rewrites_c5: !n m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2A n m)
% Assm: h4/numRing/num__rewrites_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numRing/num__rewrites_c37: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numRing/num__rewrites_c50: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/numRing/num__rewrites_c68: !n. h4/arithmetic/_2A h4/arithmetic/ZERO n = h4/arithmetic/ZERO
% Assm: h4/numRing/num__rewrites_c70: !n m. h4/arithmetic/_2A (h4/arithmetic/BIT1 n) m = h4/numeral/iZ (h4/arithmetic/_2B (h4/numeral/iDUB (h4/arithmetic/_2A n m)) m)
% Assm: h4/numRing/num__rewrites_c74: h4/numeral/iDUB h4/arithmetic/ZERO = h4/arithmetic/ZERO
% Assm: h4/numRing/num__rewrites_c75: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> 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/intExtension/INT__ABS__NOT0POS: !x. ~(x = h4/integer/int__of__num h4/num/0) ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/ABS x)
% 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/frac/FRAC__DNMPOS: !f. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/frac/frac__dnm f)
% 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/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__AINV__CALCULATE: !b1 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> h4/frac/frac__ainv (h4/frac/abs__frac (h4/pair/_2C a1 b1)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__neg a1) b1)
% Assm: h4/frac/FRAC__MINV__CALCULATE: !b1 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> ~(a1 = h4/integer/int__of__num h4/num/0) ==> h4/frac/frac__minv (h4/frac/abs__frac (h4/pair/_2C a1 b1)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/intExtension/SGN a1) b1) (h4/integer/ABS a1))
% 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/rat/rat__nmr__def: !r. h4/rat/rat__nmr r = h4/frac/frac__nmr (h4/rat/rep__rat r)
% Assm: h4/rat/rat__ainv__def: !r1. h4/rat/rat__ainv r1 = h4/rat/abs__rat (h4/frac/frac__ainv (h4/rat/rep__rat r1))
% Assm: h4/rat/rat__minv__def: !r1. h4/rat/rat__minv r1 = h4/rat/abs__rat (h4/frac/frac__minv (h4/rat/rep__rat r1))
% Assm: h4/rat/RAT: !r. h4/rat/abs__rat (h4/rat/rep__rat r) = r
% 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__EQ: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat 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/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/rat/RAT__AINV__CONG: !x. h4/rat/abs__rat (h4/frac/frac__ainv (h4/rat/rep__rat (h4/rat/abs__rat x))) = h4/rat/abs__rat (h4/frac/frac__ainv x)
% Assm: h4/rat/RAT__MINV__CONG: !x. ~(h4/frac/frac__nmr x = h4/integer/int__of__num h4/num/0) ==> h4/rat/abs__rat (h4/frac/frac__minv (h4/rat/rep__rat (h4/rat/abs__rat x))) = h4/rat/abs__rat (h4/frac/frac__minv x)
% Assm: h4/rat/RAT__AINV__CALCULATE: !f1. h4/rat/rat__ainv (h4/rat/abs__rat f1) = h4/rat/abs__rat (h4/frac/frac__ainv f1)
% 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__AINV__0: h4/rat/rat__ainv (h4/rat/rat__of__num h4/num/0) = h4/rat/rat__of__num h4/num/0
% Assm: h4/rat/RAT__AINV__AINV: !r1. h4/rat/rat__ainv (h4/rat/rat__ainv r1) = r1
% Goal: !r1. ~(r1 = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__ainv (h4/rat/rat__minv r1) = h4/rat/rat__minv (h4/rat/rat__ainv r1)
%   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_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: 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_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% 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_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_integers_INTu_u_ADDu_u_RINV]: !x. h4/integer/int__add x (h4/integer/int__neg x) = h4/integer/int__of__num h4/num/0
% 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_integers_INTu_u_LNEGu_u_UNIQ]: !y x. h4/integer/int__add x y = h4/integer/int__of__num h4/num/0 <=> x = h4/integer/int__neg y
% 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_integers_INTu_u_NEGu_u_LT0]: !x. h4/integer/int__lt (h4/integer/int__neg x) (h4/integer/int__of__num h4/num/0) <=> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) x
% 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_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_EQu_u_NEG]: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% 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_MULu_u_CALCULATEu_c1]: !y x. h4/integer/int__mul (h4/integer/int__neg x) y = h4/integer/int__neg (h4/integer/int__mul x y)
% Assm [h4s_prelims_listu_u_mergeu_u_defu_c1]: !v5 v4 a__lt. h4/prelim/list__merge a__lt h4/list/NIL (h4/list/CONS v4 v5) = h4/list/CONS v4 v5
% Assm [h4s_prelims_listu_u_mergeu_u_defu_c2]: !y x l2 l1 a__lt. h4/prelim/list__merge a__lt (h4/list/CONS x l1) (h4/list/CONS y l2) = h4/bool/COND (happ (happ a__lt x) y) (h4/list/CONS x (h4/prelim/list__merge a__lt l1 (h4/list/CONS y l2))) (h4/list/CONS y (h4/prelim/list__merge a__lt (h4/list/CONS x l1) l2))
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c1]: !v10. h4/quote/index__compare h4/quote/End__idx (h4/quote/Left__idx v10) = h4/prelim/LESS
% Assm [h4s_quotes_indexu_u_ltu_u_def]: !i2 i1. happ (happ h4/quote/index__lt i1) i2 <=> h4/quote/index__compare i1 i2 = h4/prelim/LESS
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c1]: !vm p. h4/integerRing/int__interp__p vm p = h4/integerRing/int__r__interp__cs vm (h4/integerRing/int__polynom__simplify p)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c2]: !vm c. h4/integerRing/int__interp__p vm (h4/ringNorm/Pconst c) = c
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c3]: !vm i. h4/integerRing/int__interp__p vm (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c5]: !vm p2 p1. h4/integerRing/int__interp__p vm (h4/ringNorm/Pmult p1 p2) = h4/integer/int__mul (h4/integerRing/int__interp__p vm p1) (h4/integerRing/int__interp__p vm p2)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c6]: !vm p1. h4/integerRing/int__interp__p vm (h4/ringNorm/Popp p1) = h4/integer/int__neg (h4/integerRing/int__interp__p vm p1)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c7]: !x v2 v1. h4/quote/varmap__find h4/quote/End__idx (h4/quote/Node__vm x v1 v2) = x
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c9]: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v1
% 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_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_c27]: !t l0 l c. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__monom__insert c (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_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_c30]: !t l0 l c0 c. h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__monom__insert (h4/integer/int__mul c0 c) (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c31]: !t l0 l c0. h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__monom__insert c0 (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 t)
% 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_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_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_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_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_c38]: h4/integerRing/int__r__canonical__sum__simplify h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c39]: !x vm. h4/integerRing/int__r__ivl__aux vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c40]: !x_27 x vm t_27. h4/integerRing/int__r__ivl__aux vm x (h4/list/CONS x_27 t_27) = h4/integer/int__mul (h4/quote/varmap__find x vm) (h4/integerRing/int__r__ivl__aux vm x_27 t_27)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c42]: !x vm t. h4/integerRing/int__r__interp__vl vm (h4/list/CONS x t) = h4/integerRing/int__r__ivl__aux vm x t
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c44]: !x vm t c. h4/integerRing/int__r__interp__m vm c (h4/list/CONS x t) = h4/integer/int__mul c (h4/integerRing/int__r__ivl__aux vm x t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c45]: !vm a. h4/integerRing/int__r__ics__aux vm a h4/canonical/Nil__monom = a
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c49]: !vm t l. h4/integerRing/int__r__interp__cs vm (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__vl vm l) t
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c50]: !vm t l c. h4/integerRing/int__r__interp__cs vm (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__m vm c l) t
% 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_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_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_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_c56]: !x. h4/integerRing/int__polynom__simplify x = h4/integerRing/int__r__canonical__sum__simplify (h4/integerRing/int__polynom__normalize x)
% Assm [h4s_integerRings_intu_u_rewritesu_c7]: !n m. h4/integer/int__mul (h4/integer/int__neg (h4/integer/int__of__num n)) (h4/integer/int__neg (h4/integer/int__of__num m)) = h4/integer/int__of__num (h4/arithmetic/_2A n m)
% Assm [h4s_integerRings_intu_u_rewritesu_c8]: !n m. h4/integer/int__of__num n = h4/integer/int__of__num m <=> n = m
% Assm [h4s_integerRings_intu_u_rewritesu_c10]: !n m. h4/integer/int__neg (h4/integer/int__of__num n) = h4/integer/int__of__num m <=> n = h4/num/0 /\ m = h4/num/0
% 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_c15]: h4/integer/int__1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_numRings_numu_u_rewritesu_c5]: !n m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2A n m)
% Assm [h4s_numRings_numu_u_rewritesu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numRings_numu_u_rewritesu_c37]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numRings_numu_u_rewritesu_c50]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_numRings_numu_u_rewritesu_c68]: !n. h4/arithmetic/_2A h4/arithmetic/ZERO n = h4/arithmetic/ZERO
% Assm [h4s_numRings_numu_u_rewritesu_c70]: !n m. h4/arithmetic/_2A (h4/arithmetic/BIT1 n) m = h4/numeral/iZ (h4/arithmetic/_2B (h4/numeral/iDUB (h4/arithmetic/_2A n m)) m)
% Assm [h4s_numRings_numu_u_rewritesu_c74]: h4/numeral/iDUB h4/arithmetic/ZERO = h4/arithmetic/ZERO
% Assm [h4s_numRings_numu_u_rewritesu_c75]: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> 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_intExtensions_INTu_u_ABSu_u_NOT0POS]: !x. ~(x = h4/integer/int__of__num h4/num/0) ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/ABS x)
% 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_fracs_FRACu_u_DNMPOS]: !f. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/frac/frac__dnm f)
% 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_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_AINVu_u_CALCULATE]: !b1 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> h4/frac/frac__ainv (h4/frac/abs__frac (h4/pair/_2C a1 b1)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__neg a1) b1)
% Assm [h4s_fracs_FRACu_u_MINVu_u_CALCULATE]: !b1 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> ~(a1 = h4/integer/int__of__num h4/num/0) ==> h4/frac/frac__minv (h4/frac/abs__frac (h4/pair/_2C a1 b1)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/intExtension/SGN a1) b1) (h4/integer/ABS a1))
% 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_rats_ratu_u_nmru_u_def]: !r. h4/rat/rat__nmr r = h4/frac/frac__nmr (h4/rat/rep__rat r)
% Assm [h4s_rats_ratu_u_ainvu_u_def]: !r1. h4/rat/rat__ainv r1 = h4/rat/abs__rat (h4/frac/frac__ainv (h4/rat/rep__rat r1))
% Assm [h4s_rats_ratu_u_minvu_u_def]: !r1. h4/rat/rat__minv r1 = h4/rat/abs__rat (h4/frac/frac__minv (h4/rat/rep__rat r1))
% Assm [h4s_rats_RAT]: !r. h4/rat/abs__rat (h4/rat/rep__rat r) = r
% 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_EQ]: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat 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_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_rats_RATu_u_AINVu_u_CONG]: !x. h4/rat/abs__rat (h4/frac/frac__ainv (h4/rat/rep__rat (h4/rat/abs__rat x))) = h4/rat/abs__rat (h4/frac/frac__ainv x)
% Assm [h4s_rats_RATu_u_MINVu_u_CONG]: !x. ~(h4/frac/frac__nmr x = h4/integer/int__of__num h4/num/0) ==> h4/rat/abs__rat (h4/frac/frac__minv (h4/rat/rep__rat (h4/rat/abs__rat x))) = h4/rat/abs__rat (h4/frac/frac__minv x)
% Assm [h4s_rats_RATu_u_AINVu_u_CALCULATE]: !f1. h4/rat/rat__ainv (h4/rat/abs__rat f1) = h4/rat/abs__rat (h4/frac/frac__ainv f1)
% Assm [h4s_rats_RATu_u_EQ0u_u_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 [h4s_rats_RATu_u_AINVu_u_0]: h4/rat/rat__ainv (h4/rat/rat__of__num h4/num/0) = h4/rat/rat__of__num h4/num/0
% Assm [h4s_rats_RATu_u_AINVu_u_AINV]: !r1. h4/rat/rat__ainv (h4/rat/rat__ainv r1) = r1
% Goal: !r1. ~(r1 = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__ainv (h4/rat/rat__minv r1) = h4/rat/rat__minv (h4/rat/rat__ainv r1)
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_Q93120,TV_Q93116]: ![V_f, V_g]: (![V_x]: s(TV_Q93116,happ(s(t_fun(TV_Q93120,TV_Q93116),V_f),s(TV_Q93120,V_x))) = s(TV_Q93116,happ(s(t_fun(TV_Q93120,TV_Q93116),V_g),s(TV_Q93120,V_x))) => s(t_fun(TV_Q93120,TV_Q93116),V_f) = s(t_fun(TV_Q93120,TV_Q93116),V_g))).
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_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_TRUTH, axiom, p(s(t_bool,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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_ORu_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_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_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_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_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_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_ADDu_u_RINV, 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_neg(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_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_integers_INTu_u_LNEGu_u_UNIQ, 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_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_neg(s(t_h4s_integers_int,V_y))))).
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_integers_INTu_u_NEGu_u_LT0, axiom, ![V_x]: s(t_bool,h4s_integers_intu_u_lt(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_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_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_x)))).
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_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_EQu_u_NEG, axiom, ![V_y, V_x]: (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,V_x) = s(t_h4s_integers_int,V_y))).
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_MULu_u_CALCULATEu_c1, 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,V_y))) = 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)))))).
fof(ah4s_prelims_listu_u_mergeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v5, V_v4, V_au_u_lt]: s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_v4),s(t_h4s_lists_list(TV_u_27a),V_v5))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_v4),s(t_h4s_lists_list(TV_u_27a),V_v5)))).
fof(ah4s_prelims_listu_u_mergeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_y, V_x, V_l2, V_l1, V_au_u_lt]: s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_y),s(t_h4s_lists_list(TV_u_27a),V_l2))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_y),s(t_h4s_lists_list(TV_u_27a),V_l2))))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_y),s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2)))))))).
fof(ah4s_quotes_indexu_u_compareu_u_defu_c1, axiom, ![V_v10]: s(t_h4s_prelims_ordering,h4s_quotes_indexu_u_compare(s(t_h4s_quotes_index,h4s_quotes_endu_u_idx),s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_v10))))) = s(t_h4s_prelims_ordering,h4s_prelims_less)).
fof(ah4s_quotes_indexu_u_ltu_u_def, axiom, ![V_i2, V_i1]: (p(s(t_bool,happ(s(t_fun(t_h4s_quotes_index,t_bool),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_index,V_i2)))) <=> s(t_h4s_prelims_ordering,h4s_quotes_indexu_u_compare(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_index,V_i2))) = s(t_h4s_prelims_ordering,h4s_prelims_less))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c1, axiom, ![V_vm, V_p]: s(t_h4s_integers_int,h4s_integerrings_intu_u_interpu_u_p(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p))) = 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_integerrings_intu_u_polynomu_u_simplify(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c2, axiom, ![V_vm, V_c]: s(t_h4s_integers_int,h4s_integerrings_intu_u_interpu_u_p(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pconst(s(t_h4s_integers_int,V_c))))) = s(t_h4s_integers_int,V_c)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c3, axiom, ![V_vm, V_i]: s(t_h4s_integers_int,h4s_integerrings_intu_u_interpu_u_p(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pvar(s(t_h4s_quotes_index,V_i))))) = s(t_h4s_integers_int,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c5, axiom, ![V_vm, V_p2, V_p1]: s(t_h4s_integers_int,h4s_integerrings_intu_u_interpu_u_p(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pmult(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p1),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p2))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integerrings_intu_u_interpu_u_p(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p1))),s(t_h4s_integers_int,h4s_integerrings_intu_u_interpu_u_p(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p2)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c6, axiom, ![V_vm, V_p1]: s(t_h4s_integers_int,h4s_integerrings_intu_u_interpu_u_p(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_popp(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p1))))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integerrings_intu_u_interpu_u_p(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_p1)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c7, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,h4s_quotes_endu_u_idx),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))) = s(TV_u_27a,V_x)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c9, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1, V_i1]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_varmap(TV_u_27a),V_v1)))).
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_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_c27, axiom, ![V_t, V_l0, V_l, V_c]: 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_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_integerrings_intu_u_ru_u_monomu_u_insert(s(t_h4s_integers_int,V_c),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_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_c30, axiom, ![V_t, V_l0, V_l, V_c0, V_c]: 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_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_integerrings_intu_u_ru_u_monomu_u_insert(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_c0),s(t_h4s_integers_int,V_c))),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_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),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c31, axiom, ![V_t, V_l0, V_l, 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_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_monomu_u_insert(s(t_h4s_integers_int,V_c0),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_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),V_t)))))).
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_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_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_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_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_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_c39, axiom, ![V_x, V_vm]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(t_h4s_integers_int,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c40, axiom, ![V_xu_27, V_x, V_vm, V_tu_27]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm))),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c42, axiom, ![V_x, V_vm, V_t]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_vl(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t))))) = s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c44, axiom, ![V_x, V_vm, V_t, V_c]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_m(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_c),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c45, axiom, ![V_vm, V_a]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_icsu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,V_a),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_integers_int,V_a)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c49, axiom, ![V_vm, V_t, V_l]: 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_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_integers_int,h4s_integerrings_intu_u_ru_u_icsu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_vl(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c50, axiom, ![V_vm, V_t, V_l, V_c]: 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_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_integers_int,h4s_integerrings_intu_u_ru_u_icsu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_m(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),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)))).
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_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_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_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_c56, axiom, ![V_x]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_simplify(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_x))) = 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_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_x)))))).
fof(ah4s_integerRings_intu_u_rewritesu_c7, axiom, ![V_n, V_m]: 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,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_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_integerRings_intu_u_rewritesu_c8, axiom, ![V_n, 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,V_m))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m))).
fof(ah4s_integerRings_intu_u_rewritesu_c10, axiom, ![V_n, 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,V_n))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) <=> (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0) & s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0)))).
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_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_numRings_numu_u_rewritesu_c5, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_numRings_numu_u_rewritesu_c17, axiom, ![V_n]: (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_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numRings_numu_u_rewritesu_c37, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_numRings_numu_u_rewritesu_c50, 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,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numRings_numu_u_rewritesu_c68, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero)).
fof(ah4s_numRings_numu_u_rewritesu_c70, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,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_numerals_idub(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_numRings_numu_u_rewritesu_c74, axiom, s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_h4s_nums_num,h4s_arithmetics_zero)).
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_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_intExtensions_INTu_u_ABSu_u_NOT0POS, axiom, ![V_x]: (~ (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,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_x)))))))).
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_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_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_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_AINVu_u_CALCULATE, axiom, ![V_b1, 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)))) => s(t_h4s_fracs_frac,h4s_fracs_fracu_u_ainv(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,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_a1))),s(t_h4s_integers_int,V_b1))))))).
fof(ah4s_fracs_FRACu_u_MINVu_u_CALCULATE, axiom, ![V_b1, 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)))) => (~ (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_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,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,V_b1))),s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_a1)))))))))).
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_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_rats_ratu_u_ainvu_u_def, axiom, ![V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_ainv(s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,V_r1)))))))).
fof(ah4s_rats_ratu_u_minvu_u_def, axiom, ![V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,V_r1))) = 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,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,V_r1)))))))).
fof(ah4s_rats_RAT, axiom, ![V_r]: s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(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_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_EQ, 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))) <=> 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_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_rats_RATu_u_AINVu_u_CONG, axiom, ![V_x]: s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_ainv(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_x))))))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_ainv(s(t_h4s_fracs_frac,V_x)))))).
fof(ah4s_rats_RATu_u_MINVu_u_CONG, axiom, ![V_x]: (~ (s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,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_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_minv(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_x))))))))) = 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_x))))))).
fof(ah4s_rats_RATu_u_AINVu_u_CALCULATE, axiom, ![V_f1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(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_ainv(s(t_h4s_fracs_frac,V_f1)))))).
fof(ah4s_rats_RATu_u_EQ0u_u_NMR, axiom, ![V_r1]: (s(t_h4s_rats_rat,V_r1) = s(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_AINVu_u_0, axiom, s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(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_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_rats_RATu_u_AINVu_u_AINV, axiom, ![V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r1))))) = s(t_h4s_rats_rat,V_r1)).
fof(ch4s_rats_RATu_u_AINVu_u_MINV, conjecture, ![V_r1]: (~ (s(t_h4s_rats_rat,V_r1) = s(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_ainv(s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,V_r1))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r1))))))).
