%   ORIGINAL: h4/rat/RAT__MUL__CONG1
% 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/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% 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_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~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/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/NOT__NOT: !t. ~ ~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/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% 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/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/quotient/QUOTIENT__def: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. abs (rep a) = a) /\ (!a. R (rep a) (rep a)) /\ (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/integer/INT__EQ__RMUL: !z y x. h4/integer/int__mul x z = h4/integer/int__mul y z <=> z = h4/integer/int__of__num h4/num/0 \/ x = y
% Assm: h4/integer/INT__INJ: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm: h4/prelim/ordering__eq__dec_c5: h4/prelim/GREATER = h4/prelim/LESS <=> F
% Assm: h4/prelim/list__merge__def_c0: !l1 a__lt. h4/prelim/list__merge a__lt l1 h4/list/NIL = l1
% 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__compare__def_c2: !v11. h4/quote/index__compare h4/quote/End__idx (h4/quote/Right__idx v11) = h4/prelim/LESS
% Assm: h4/quote/index__compare__def_c5: !n_27 m_27. h4/quote/index__compare (h4/quote/Left__idx n_27) (h4/quote/Left__idx m_27) = h4/quote/index__compare n_27 m_27
% Assm: h4/quote/index__compare__def_c8: !n_27 m_27. h4/quote/index__compare (h4/quote/Right__idx n_27) (h4/quote/Left__idx m_27) = h4/prelim/GREATER
% 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_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_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_c8: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Right__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v2
% 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_c23: !l1. h4/integerRing/int__r__varlist__insert l1 h4/canonical/Nil__monom = h4/canonical/Cons__varlist l1 h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c28: !t l0 l. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar2 l0 t)
% Assm: h4/integerRing/int__ring__thms_c29: !l0. h4/integerRing/int__r__canonical__sum__scalar2 l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c34: !t1 s2 l1. h4/integerRing/int__r__canonical__sum__prod (h4/canonical/Cons__varlist l1 t1) s2 = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__r__canonical__sum__scalar2 l1 s2) (h4/integerRing/int__r__canonical__sum__prod t1 s2)
% Assm: h4/integerRing/int__ring__thms_c35: !s2. h4/integerRing/int__r__canonical__sum__prod h4/canonical/Nil__monom s2 = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c37: !t l. h4/integerRing/int__r__canonical__sum__simplify (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__varlist 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_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_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_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_c56: !x. h4/integerRing/int__polynom__simplify x = h4/integerRing/int__r__canonical__sum__simplify (h4/integerRing/int__polynom__normalize x)
% Assm: h4/numRing/num__rewrites_c75: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> T
% Assm: h4/intExtension/INT__MUL__POS__SIGN: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) a ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul a b)
% Assm: h4/intExtension/LESS__IMP__NOT__0: !n. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) n ==> ~(n = h4/integer/int__of__num h4/num/0)
% Assm: h4/frac/frac__mul__def: !f2 f1. h4/frac/frac__mul f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__nmr f2)) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% 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/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__EQUIV__SYM: !b a. h4/rat/rat__equiv a b <=> h4/rat/rat__equiv b a
% Assm: h4/rat/rat__QUOTIENT: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm: h4/rat/RAT__ABS__EQUIV: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat f2 <=> h4/rat/rat__equiv f1 f2
% Goal: !y x. h4/rat/abs__rat (h4/frac/frac__mul (h4/rat/rep__rat (h4/rat/abs__rat x)) y) = h4/rat/abs__rat (h4/frac/frac__mul x y)
%   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_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% 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_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~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_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~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_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% 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_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_quotients_QUOTIENTu_u_def]: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. happ abs (happ rep a) = a) /\ (!a. happ (happ R (happ rep a)) (happ rep a)) /\ (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_integers_INTu_u_EQu_u_RMUL]: !z y x. h4/integer/int__mul x z = h4/integer/int__mul y z <=> z = h4/integer/int__of__num h4/num/0 \/ x = y
% Assm [h4s_integers_INTu_u_INJ]: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm [h4s_prelims_orderingu_u_equ_u_decu_c5]: h4/prelim/GREATER = h4/prelim/LESS <=> F
% Assm [h4s_prelims_listu_u_mergeu_u_defu_c0]: !l1 a__lt. h4/prelim/list__merge a__lt l1 h4/list/NIL = l1
% 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_compareu_u_defu_c2]: !v11. h4/quote/index__compare h4/quote/End__idx (h4/quote/Right__idx v11) = h4/prelim/LESS
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c5]: !n_27 m_27. h4/quote/index__compare (h4/quote/Left__idx n_27) (h4/quote/Left__idx m_27) = h4/quote/index__compare n_27 m_27
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c8]: !n_27 m_27. h4/quote/index__compare (h4/quote/Right__idx n_27) (h4/quote/Left__idx m_27) = h4/prelim/GREATER
% 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_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_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_c8]: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Right__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v2
% 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_c23]: !l1. h4/integerRing/int__r__varlist__insert l1 h4/canonical/Nil__monom = h4/canonical/Cons__varlist l1 h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c28]: !t l0 l. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar2 l0 t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_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_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_c37]: !t l. h4/integerRing/int__r__canonical__sum__simplify (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__varlist 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_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_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_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_c56]: !x. h4/integerRing/int__polynom__simplify x = h4/integerRing/int__r__canonical__sum__simplify (h4/integerRing/int__polynom__normalize x)
% Assm [h4s_numRings_numu_u_rewritesu_c75]: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> T
% Assm [h4s_intExtensions_INTu_u_MULu_u_POSu_u_SIGN]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) a ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul a b)
% Assm [h4s_intExtensions_LESSu_u_IMPu_u_NOTu_u_0]: !n. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) n ==> ~(n = h4/integer/int__of__num h4/num/0)
% Assm [h4s_fracs_fracu_u_mulu_u_def]: !f2 f1. h4/frac/frac__mul f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__nmr f2)) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% 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_rats_ratu_u_equivu_u_def]: !f2 f1. happ (happ h4/rat/rat__equiv f1) f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm [h4s_rats_RATu_u_EQUIVu_u_SYM]: !b a. happ (happ h4/rat/rat__equiv a) b <=> happ (happ h4/rat/rat__equiv b) a
% Assm [h4s_rats_ratu_u_QUOTIENT]: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm [h4s_rats_RATu_u_ABSu_u_EQUIV]: !f2 f1. happ h4/rat/abs__rat f1 = happ h4/rat/abs__rat f2 <=> happ (happ h4/rat/rat__equiv f1) f2
% Goal: !y x. happ h4/rat/abs__rat (h4/frac/frac__mul (happ h4/rat/rep__rat (happ h4/rat/abs__rat x)) y) = happ h4/rat/abs__rat (h4/frac/frac__mul x y)
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_Q91813,TV_Q91809]: ![V_f, V_g]: (![V_x]: s(TV_Q91809,happ(s(t_fun(TV_Q91813,TV_Q91809),V_f),s(TV_Q91813,V_x))) = s(TV_Q91809,happ(s(t_fun(TV_Q91813,TV_Q91809),V_g),s(TV_Q91813,V_x))) => s(t_fun(TV_Q91813,TV_Q91809),V_f) = s(t_fun(TV_Q91813,TV_Q91809),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_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_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_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_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_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_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_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
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_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_NOTu_u_FORALLu_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_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (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_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_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_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_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_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_quotients_QUOTIENTu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & (![V_a]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))) & ![V_r, V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_s)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_r)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))))).
fof(ah4s_integers_INTu_u_EQu_u_RMUL, 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_z))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))) <=> (s(t_h4s_integers_int,V_z) = 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,V_y)))).
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_prelims_orderingu_u_equ_u_decu_c5, axiom, (s(t_h4s_prelims_ordering,h4s_prelims_greater) = s(t_h4s_prelims_ordering,h4s_prelims_less) <=> p(s(t_bool,f)))).
fof(ah4s_prelims_listu_u_mergeu_u_defu_c0, axiom, ![TV_u_27a]: ![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),V_l1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),V_l1)).
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_compareu_u_defu_c2, axiom, ![V_v11]: 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_rightu_u_idx(s(t_h4s_quotes_index,V_v11))))) = s(t_h4s_prelims_ordering,h4s_prelims_less)).
fof(ah4s_quotes_indexu_u_compareu_u_defu_c5, axiom, ![V_nu_27, V_mu_27]: s(t_h4s_prelims_ordering,h4s_quotes_indexu_u_compare(s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_nu_27))),s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_mu_27))))) = s(t_h4s_prelims_ordering,h4s_quotes_indexu_u_compare(s(t_h4s_quotes_index,V_nu_27),s(t_h4s_quotes_index,V_mu_27)))).
fof(ah4s_quotes_indexu_u_compareu_u_defu_c8, axiom, ![V_nu_27, V_mu_27]: s(t_h4s_prelims_ordering,h4s_quotes_indexu_u_compare(s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_nu_27))),s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_mu_27))))) = s(t_h4s_prelims_ordering,h4s_prelims_greater)).
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_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_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_c8, 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_rightu_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_v2)))).
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_c23, axiom, ![V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_varlistu_u_insert(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c28, axiom, ![V_t, V_l0, V_l]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_varlistu_u_insert(s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_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_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_c37, axiom, ![V_t, 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),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_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)))))).
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_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_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_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_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_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_INTu_u_MULu_u_POSu_u_SIGN, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_a)))) => (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))))).
fof(ah4s_intExtensions_LESSu_u_IMPu_u_NOTu_u_0, axiom, ![V_n]: (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_n)))) => ~ (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)))))).
fof(ah4s_fracs_fracu_u_mulu_u_def, axiom, ![V_f2, V_f1]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_mul(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_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_nmr(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_dnm(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2)))))))))).
fof(ah4s_fracs_FRACu_u_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_rats_ratu_u_equivu_u_def, axiom, ![V_f2, V_f1]: (p(s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_f1))),s(t_h4s_fracs_frac,V_f2)))) <=> s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))))))).
fof(ah4s_rats_RATu_u_EQUIVu_u_SYM, axiom, ![V_b, V_a]: s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_a))),s(t_h4s_fracs_frac,V_b))) = s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_b))),s(t_h4s_fracs_frac,V_a)))).
fof(ah4s_rats_ratu_u_QUOTIENT, axiom, p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat))))).
fof(ah4s_rats_RATu_u_ABSu_u_EQUIV, axiom, ![V_f2, V_f1]: (s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,V_f1))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,V_f2))) <=> p(s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_f1))),s(t_h4s_fracs_frac,V_f2)))))).
fof(ch4s_rats_RATu_u_MULu_u_CONG1, conjecture, ![V_y, V_x]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_mul(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,V_x))))),s(t_h4s_fracs_frac,V_y))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_mul(s(t_h4s_fracs_frac,V_x),s(t_h4s_fracs_frac,V_y)))))).
