%   ORIGINAL: h4/rat/RAT__LES__TRANS
% 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/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/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> 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/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/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/integer/int__sub0: !y x. h4/integer/int__sub x y = h4/integer/int__add x (h4/integer/int__neg y)
% Assm: h4/integer/int__gt0: !y x. h4/integer/int__gt x y <=> h4/integer/int__lt y x
% Assm: h4/integer/INT__LT__ADD: !y x. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) x /\ h4/integer/int__lt (h4/integer/int__of__num h4/num/0) y ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__add x y)
% Assm: h4/prelim/compare__def_c0: !lt gt eq. h4/prelim/compare h4/prelim/LESS lt eq gt = lt
% Assm: h4/prelim/compare__def_c1: !lt gt eq. h4/prelim/compare h4/prelim/EQUAL lt eq gt = eq
% Assm: h4/prelim/ordering__eq__dec_c4: h4/prelim/EQUAL = h4/prelim/LESS <=> F
% Assm: h4/prelim/ordering__eq__dec_c5: h4/prelim/GREATER = h4/prelim/LESS <=> F
% Assm: h4/prelim/list__compare__def_c0: !cmp. h4/prelim/list__compare cmp h4/list/NIL h4/list/NIL = h4/prelim/EQUAL
% Assm: h4/prelim/list__compare__def_c3: !y x l2 l1 cmp. h4/prelim/list__compare cmp (h4/list/CONS x l1) (h4/list/CONS y l2) = h4/prelim/compare (cmp x y) h4/prelim/LESS (h4/prelim/list__compare cmp l1 l2) h4/prelim/GREATER
% 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_c0: h4/quote/index__compare h4/quote/End__idx h4/quote/End__idx = h4/prelim/EQUAL
% 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_c3: !v2. h4/quote/index__compare (h4/quote/Left__idx v2) h4/quote/End__idx = h4/prelim/GREATER
% Assm: h4/quote/index__compare__def_c4: !v3. h4/quote/index__compare (h4/quote/Right__idx v3) h4/quote/End__idx = h4/prelim/GREATER
% 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_c6: !n_27 m_27. h4/quote/index__compare (h4/quote/Left__idx n_27) (h4/quote/Right__idx m_27) = h4/prelim/LESS
% Assm: h4/quote/index__compare__def_c7: !n_27 m_27. h4/quote/index__compare (h4/quote/Right__idx n_27) (h4/quote/Right__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_c4: !vm p2 p1. h4/integerRing/int__interp__p vm (h4/ringNorm/Pplus p1 p2) = h4/integer/int__add (h4/integerRing/int__interp__p vm p1) (h4/integerRing/int__interp__p vm p2)
% 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_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_c11: !t2 t1 l2 l1 c2 c1. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add c1 c2) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/integerRing/int__ring__thms_c12: !t2 t1 l2 l1 c1. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add c1 h4/integer/int__1) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__varlist l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/integerRing/int__ring__thms_c13: !t2 t1 l2 l1 c2. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add h4/integer/int__1 c2) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/integerRing/int__ring__thms_c14: !t2 t1 l2 l1. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add h4/integer/int__1 h4/integer/int__1) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__varlist l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% 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_c16: !v6 v5 v4. h4/integerRing/int__r__canonical__sum__merge h4/canonical/Nil__monom (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% 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_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_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_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_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_c47: !vm t l c a. h4/integerRing/int__r__ics__aux vm a (h4/canonical/Cons__monom c l t) = h4/integer/int__add a (h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__m vm c l) t)
% 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_c53: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pplus pl pr) = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm: h4/integerRing/int__ring__thms_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_c1: !n m. h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num n)) (h4/integer/int__of__num m) = h4/bool/COND (h4/arithmetic/_3C_3D n m) (h4/integer/int__of__num (h4/arithmetic/_2D m n)) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/_2D 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/integerRing/int__rewrites_c18: !n. h4/prim__rec/_3C n h4/arithmetic/ZERO <=> F
% Assm: h4/integerRing/int__rewrites_c19: !n m. h4/prim__rec/_3C (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/prim__rec/_3C n m
% Assm: h4/integerRing/int__rewrites_c23: !n. h4/arithmetic/_3C_3D h4/arithmetic/ZERO n <=> T
% Assm: h4/integerRing/int__rewrites_c26: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/integerRing/int__rewrites_c30: !n m. h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m) = h4/bool/COND (h4/prim__rec/_3C m n) (h4/arithmetic/NUMERAL (h4/numeral/iSUB T n m)) h4/num/0
% Assm: h4/numRing/num__rewrites_c8: !n m. h4/arithmetic/_2D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m)
% Assm: h4/numRing/num__rewrites_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numRing/num__rewrites_c28: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/numRing/num__rewrites_c37: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% 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/INT__SGN__CLAUSES_c2: !x. h4/intExtension/SGN x = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) <=> h4/integer/int__gt x (h4/integer/int__of__num h4/num/0)
% Assm: h4/frac/frac__sgn__def: !f1. h4/frac/frac__sgn f1 = h4/intExtension/SGN (h4/frac/frac__nmr f1)
% 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__ADD__CALCULATE: !b2 b1 a2 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b2 ==> h4/frac/frac__add (h4/frac/abs__frac (h4/pair/_2C a1 b1)) (h4/frac/abs__frac (h4/pair/_2C a2 b2)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__add (h4/integer/int__mul a1 b2) (h4/integer/int__mul a2 b1)) (h4/integer/int__mul b1 b2))
% Assm: h4/frac/FRAC__SUB__CALCULATE: !b2 b1 a2 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b2 ==> h4/frac/frac__sub (h4/frac/abs__frac (h4/pair/_2C a1 b1)) (h4/frac/abs__frac (h4/pair/_2C a2 b2)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__sub (h4/integer/int__mul a1 b2) (h4/integer/int__mul a2 b1)) (h4/integer/int__mul b1 b2))
% Assm: h4/rat/rat__sgn__def: !r. h4/rat/rat__sgn r = h4/frac/frac__sgn (h4/rat/rep__rat r)
% Assm: h4/rat/rat__add__def: !r2 r1. h4/rat/rat__add r1 r2 = h4/rat/abs__rat (h4/frac/frac__add (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm: h4/rat/rat__sub__def: !r2 r1. h4/rat/rat__sub r1 r2 = h4/rat/abs__rat (h4/frac/frac__sub (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm: h4/rat/rat__les__def: !r2 r1. h4/rat/rat__les r1 r2 <=> h4/rat/rat__sgn (h4/rat/rat__sub r2 r1) = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/rat/RAT: !r. h4/rat/abs__rat (h4/rat/rep__rat r) = r
% 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__SGN__CONG: !f1. h4/frac/frac__sgn (h4/rat/rep__rat (h4/rat/abs__rat f1)) = h4/frac/frac__sgn f1
% Assm: h4/rat/RAT__ADD__CONG_c0: !y x. h4/rat/abs__rat (h4/frac/frac__add (h4/rat/rep__rat (h4/rat/abs__rat x)) y) = h4/rat/abs__rat (h4/frac/frac__add x y)
% Assm: h4/rat/RAT__ADD__CONG_c1: !y x. h4/rat/abs__rat (h4/frac/frac__add x (h4/rat/rep__rat (h4/rat/abs__rat y))) = h4/rat/abs__rat (h4/frac/frac__add x y)
% Assm: h4/rat/RAT__ADD__CALCULATE: !f2 f1. h4/rat/rat__add (h4/rat/abs__rat f1) (h4/rat/abs__rat f2) = h4/rat/abs__rat (h4/frac/frac__add f1 f2)
% Assm: h4/rat/RAT__SUB__CALCULATE: !f2 f1. h4/rat/rat__sub (h4/rat/abs__rat f1) (h4/rat/abs__rat f2) = h4/rat/abs__rat (h4/frac/frac__sub f1 f2)
% Goal: !r3 r2 r1. h4/rat/rat__les r1 r2 /\ h4/rat/rat__les r2 r3 ==> h4/rat/rat__les r1 r3
%   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_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_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> 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_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_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_integers_intu_u_sub0]: !y x. h4/integer/int__sub x y = h4/integer/int__add x (h4/integer/int__neg y)
% Assm [h4s_integers_intu_u_gt0]: !y x. h4/integer/int__gt x y <=> h4/integer/int__lt y x
% Assm [h4s_integers_INTu_u_LTu_u_ADD]: !y x. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) x /\ h4/integer/int__lt (h4/integer/int__of__num h4/num/0) y ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__add x y)
% Assm [h4s_prelims_compareu_u_defu_c0]: !lt gt eq. h4/prelim/compare h4/prelim/LESS lt eq gt = lt
% Assm [h4s_prelims_compareu_u_defu_c1]: !lt gt eq. h4/prelim/compare h4/prelim/EQUAL lt eq gt = eq
% Assm [h4s_prelims_orderingu_u_equ_u_decu_c4]: h4/prelim/EQUAL = h4/prelim/LESS <=> F
% Assm [h4s_prelims_orderingu_u_equ_u_decu_c5]: h4/prelim/GREATER = h4/prelim/LESS <=> F
% Assm [h4s_prelims_listu_u_compareu_u_defu_c0]: !cmp. h4/prelim/list__compare cmp h4/list/NIL h4/list/NIL = h4/prelim/EQUAL
% Assm [h4s_prelims_listu_u_compareu_u_defu_c3]: !y x l2 l1 cmp. h4/prelim/list__compare cmp (h4/list/CONS x l1) (h4/list/CONS y l2) = h4/prelim/compare (happ (happ cmp x) y) h4/prelim/LESS (h4/prelim/list__compare cmp l1 l2) h4/prelim/GREATER
% 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_c0]: happ (happ h4/quote/index__compare h4/quote/End__idx) h4/quote/End__idx = h4/prelim/EQUAL
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c1]: !v10. happ (happ 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. happ (happ h4/quote/index__compare h4/quote/End__idx) (h4/quote/Right__idx v11) = h4/prelim/LESS
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c3]: !v2. happ (happ h4/quote/index__compare (h4/quote/Left__idx v2)) h4/quote/End__idx = h4/prelim/GREATER
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c4]: !v3. happ (happ h4/quote/index__compare (h4/quote/Right__idx v3)) h4/quote/End__idx = h4/prelim/GREATER
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c5]: !n_27 m_27. happ (happ h4/quote/index__compare (h4/quote/Left__idx n_27)) (h4/quote/Left__idx m_27) = happ (happ h4/quote/index__compare n_27) m_27
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c6]: !n_27 m_27. happ (happ h4/quote/index__compare (h4/quote/Left__idx n_27)) (h4/quote/Right__idx m_27) = h4/prelim/LESS
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c7]: !n_27 m_27. happ (happ h4/quote/index__compare (h4/quote/Right__idx n_27)) (h4/quote/Right__idx m_27) = happ (happ h4/quote/index__compare n_27) m_27
% Assm [h4s_quotes_indexu_u_compareu_u_defu_c8]: !n_27 m_27. happ (happ 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 <=> happ (happ 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_c4]: !vm p2 p1. h4/integerRing/int__interp__p vm (h4/ringNorm/Pplus p1 p2) = h4/integer/int__add (h4/integerRing/int__interp__p vm p1) (h4/integerRing/int__interp__p vm p2)
% 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_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_c11]: !t2 t1 l2 l1 c2 c1. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add c1 c2) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c12]: !t2 t1 l2 l1 c1. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add c1 h4/integer/int__1) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__varlist l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c13]: !t2 t1 l2 l1 c2. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add h4/integer/int__1 c2) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c14]: !t2 t1 l2 l1. h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/integerRing/int__r__canonical__sum__merge t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/integer/int__add h4/integer/int__1 h4/integer/int__1) l1 (h4/integerRing/int__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__varlist l2 (h4/integerRing/int__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% 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_c16]: !v6 v5 v4. h4/integerRing/int__r__canonical__sum__merge h4/canonical/Nil__monom (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% 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_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_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_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_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_c47]: !vm t l c a. h4/integerRing/int__r__ics__aux vm a (h4/canonical/Cons__monom c l t) = h4/integer/int__add a (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_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_c53]: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pplus pl pr) = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_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_c1]: !n m. h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num n)) (h4/integer/int__of__num m) = h4/bool/COND (h4/arithmetic/_3C_3D n m) (h4/integer/int__of__num (h4/arithmetic/_2D m n)) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/_2D 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_integerRings_intu_u_rewritesu_c18]: !n. h4/prim__rec/_3C n h4/arithmetic/ZERO <=> F
% Assm [h4s_integerRings_intu_u_rewritesu_c19]: !n m. h4/prim__rec/_3C (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/prim__rec/_3C n m
% Assm [h4s_integerRings_intu_u_rewritesu_c23]: !n. h4/arithmetic/_3C_3D h4/arithmetic/ZERO n <=> T
% Assm [h4s_integerRings_intu_u_rewritesu_c26]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_integerRings_intu_u_rewritesu_c30]: !n m. h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m) = h4/bool/COND (h4/prim__rec/_3C m n) (h4/arithmetic/NUMERAL (h4/numeral/iSUB T n m)) h4/num/0
% Assm [h4s_numRings_numu_u_rewritesu_c8]: !n m. h4/arithmetic/_2D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m)
% Assm [h4s_numRings_numu_u_rewritesu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numRings_numu_u_rewritesu_c28]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_numRings_numu_u_rewritesu_c37]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% 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_INTu_u_SGNu_u_CLAUSESu_c2]: !x. h4/intExtension/SGN x = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) <=> h4/integer/int__gt x (h4/integer/int__of__num h4/num/0)
% Assm [h4s_fracs_fracu_u_sgnu_u_def]: !f1. h4/frac/frac__sgn f1 = h4/intExtension/SGN (h4/frac/frac__nmr f1)
% 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_ADDu_u_CALCULATE]: !b2 b1 a2 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b2 ==> h4/frac/frac__add (h4/frac/abs__frac (h4/pair/_2C a1 b1)) (h4/frac/abs__frac (h4/pair/_2C a2 b2)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__add (h4/integer/int__mul a1 b2) (h4/integer/int__mul a2 b1)) (h4/integer/int__mul b1 b2))
% Assm [h4s_fracs_FRACu_u_SUBu_u_CALCULATE]: !b2 b1 a2 a1. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b1 ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b2 ==> h4/frac/frac__sub (h4/frac/abs__frac (h4/pair/_2C a1 b1)) (h4/frac/abs__frac (h4/pair/_2C a2 b2)) = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__sub (h4/integer/int__mul a1 b2) (h4/integer/int__mul a2 b1)) (h4/integer/int__mul b1 b2))
% Assm [h4s_rats_ratu_u_sgnu_u_def]: !r. h4/rat/rat__sgn r = h4/frac/frac__sgn (h4/rat/rep__rat r)
% Assm [h4s_rats_ratu_u_addu_u_def]: !r2 r1. h4/rat/rat__add r1 r2 = h4/rat/abs__rat (h4/frac/frac__add (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm [h4s_rats_ratu_u_subu_u_def]: !r2 r1. h4/rat/rat__sub r1 r2 = h4/rat/abs__rat (h4/frac/frac__sub (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm [h4s_rats_ratu_u_lesu_u_def]: !r2 r1. h4/rat/rat__les r1 r2 <=> h4/rat/rat__sgn (h4/rat/rat__sub r2 r1) = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_rats_RAT]: !r. h4/rat/abs__rat (h4/rat/rep__rat r) = r
% 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_SGNu_u_CONG]: !f1. h4/frac/frac__sgn (h4/rat/rep__rat (h4/rat/abs__rat f1)) = h4/frac/frac__sgn f1
% Assm [h4s_rats_RATu_u_ADDu_u_CONGu_c0]: !y x. h4/rat/abs__rat (h4/frac/frac__add (h4/rat/rep__rat (h4/rat/abs__rat x)) y) = h4/rat/abs__rat (h4/frac/frac__add x y)
% Assm [h4s_rats_RATu_u_ADDu_u_CONGu_c1]: !y x. h4/rat/abs__rat (h4/frac/frac__add x (h4/rat/rep__rat (h4/rat/abs__rat y))) = h4/rat/abs__rat (h4/frac/frac__add x y)
% Assm [h4s_rats_RATu_u_ADDu_u_CALCULATE]: !f2 f1. h4/rat/rat__add (h4/rat/abs__rat f1) (h4/rat/abs__rat f2) = h4/rat/abs__rat (h4/frac/frac__add f1 f2)
% Assm [h4s_rats_RATu_u_SUBu_u_CALCULATE]: !f2 f1. h4/rat/rat__sub (h4/rat/abs__rat f1) (h4/rat/abs__rat f2) = h4/rat/abs__rat (h4/frac/frac__sub f1 f2)
% Goal: !r3 r2 r1. h4/rat/rat__les r1 r2 /\ h4/rat/rat__les r2 r3 ==> h4/rat/rat__les r1 r3
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_Q93676,TV_Q93672]: ![V_f, V_g]: (![V_x]: s(TV_Q93672,happ(s(t_fun(TV_Q93676,TV_Q93672),V_f),s(TV_Q93676,V_x))) = s(TV_Q93672,happ(s(t_fun(TV_Q93676,TV_Q93672),V_g),s(TV_Q93676,V_x))) => s(t_fun(TV_Q93676,TV_Q93672),V_f) = s(t_fun(TV_Q93676,TV_Q93672),V_g))).
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_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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
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_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_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_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_integers_intu_u_sub0, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_sub(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y)))))).
fof(ah4s_integers_intu_u_gt0, axiom, ![V_y, V_x]: s(t_bool,h4s_integers_intu_u_gt(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x)))).
fof(ah4s_integers_INTu_u_LTu_u_ADD, axiom, ![V_y, V_x]: ((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_x)))) & 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_y))))) => 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_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y)))))))).
fof(ah4s_prelims_compareu_u_defu_c0, axiom, ![TV_u_27a]: ![V_lt, V_gt, V_eq]: s(TV_u_27a,h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_less),s(TV_u_27a,V_lt),s(TV_u_27a,V_eq),s(TV_u_27a,V_gt))) = s(TV_u_27a,V_lt)).
fof(ah4s_prelims_compareu_u_defu_c1, axiom, ![TV_u_27a]: ![V_lt, V_gt, V_eq]: s(TV_u_27a,h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_equal),s(TV_u_27a,V_lt),s(TV_u_27a,V_eq),s(TV_u_27a,V_gt))) = s(TV_u_27a,V_eq)).
fof(ah4s_prelims_orderingu_u_equ_u_decu_c4, axiom, (s(t_h4s_prelims_ordering,h4s_prelims_equal) = s(t_h4s_prelims_ordering,h4s_prelims_less) <=> p(s(t_bool,f)))).
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_compareu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_cmp]: s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_h4s_prelims_ordering,h4s_prelims_equal)).
fof(ah4s_prelims_listu_u_compareu_u_defu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_l2, V_l1, V_cmp]: s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),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_27b),h4s_lists_cons(s(TV_u_27b,V_y),s(t_h4s_lists_list(TV_u_27b),V_l2))))) = s(t_h4s_prelims_ordering,h4s_prelims_compare(s(t_h4s_prelims_ordering,happ(s(t_fun(TV_u_27b,t_h4s_prelims_ordering),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_h4s_prelims_ordering,h4s_prelims_less),s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))),s(t_h4s_prelims_ordering,h4s_prelims_greater)))).
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_c0, axiom, s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_quotes_index,h4s_quotes_endu_u_idx))),s(t_h4s_quotes_index,h4s_quotes_endu_u_idx))) = s(t_h4s_prelims_ordering,h4s_prelims_equal)).
fof(ah4s_quotes_indexu_u_compareu_u_defu_c1, axiom, ![V_v10]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_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,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_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_c3, axiom, ![V_v2]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_v2))))),s(t_h4s_quotes_index,h4s_quotes_endu_u_idx))) = s(t_h4s_prelims_ordering,h4s_prelims_greater)).
fof(ah4s_quotes_indexu_u_compareu_u_defu_c4, axiom, ![V_v3]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_v3))))),s(t_h4s_quotes_index,h4s_quotes_endu_u_idx))) = s(t_h4s_prelims_ordering,h4s_prelims_greater)).
fof(ah4s_quotes_indexu_u_compareu_u_defu_c5, axiom, ![V_nu_27, V_mu_27]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_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,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_quotes_index,V_nu_27))),s(t_h4s_quotes_index,V_mu_27)))).
fof(ah4s_quotes_indexu_u_compareu_u_defu_c6, axiom, ![V_nu_27, V_mu_27]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_nu_27))))),s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_mu_27))))) = s(t_h4s_prelims_ordering,h4s_prelims_less)).
fof(ah4s_quotes_indexu_u_compareu_u_defu_c7, axiom, ![V_nu_27, V_mu_27]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_nu_27))))),s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_mu_27))))) = s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_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,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_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,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_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_c4, 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_pplus(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_add(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_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_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_c11, axiom, ![V_t2, V_t1, V_l2, V_l1, V_c2, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_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),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_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_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_c1),s(t_h4s_integers_int,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_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_t2)))))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c12, axiom, ![V_t2, V_t1, V_l2, 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_merge(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),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_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_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_c1),s(t_h4s_integers_int,h4s_integers_intu_u_1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_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_t2)))))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c13, axiom, ![V_t2, V_t1, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_h4s_integers_int,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2)))))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c14, axiom, ![V_t2, V_t1, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_h4s_integers_int,h4s_integers_intu_u_1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t2)))))))).
fof(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_c16, axiom, ![V_v6, V_v5, V_v4]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_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_v4),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_v6))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_v4),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_v6)))).
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_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_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_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_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_c47, axiom, ![V_vm, V_t, V_l, V_c, 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_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_integers_intu_u_add(s(t_h4s_integers_int,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,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_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_c53, axiom, ![V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pplus(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_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_c1, axiom, ![V_n, V_m]: s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) = s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),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_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_u_2d(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_integerRings_intu_u_rewritesu_c18, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_integerRings_intu_u_rewritesu_c19, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_integerRings_intu_u_rewritesu_c23, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))) = s(t_bool,t)).
fof(ah4s_integerRings_intu_u_rewritesu_c26, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_integerRings_intu_u_rewritesu_c30, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_numRings_numu_u_rewritesu_c8, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_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_c28, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_numRings_numu_u_rewritesu_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_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_INTu_u_SGNu_u_CLAUSESu_c2, axiom, ![V_x]: (s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) <=> p(s(t_bool,h4s_integers_intu_u_gt(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_fracs_fracu_u_sgnu_u_def, axiom, ![V_f1]: s(t_h4s_integers_int,h4s_fracs_fracu_u_sgn(s(t_h4s_fracs_frac,V_f1))) = s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1)))))).
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_ADDu_u_CALCULATE, axiom, ![V_b2, V_b1, V_a2, V_a1]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b1)))) => (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b2)))) => s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a1),s(t_h4s_integers_int,V_b1))))),s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a2),s(t_h4s_integers_int,V_b2))))))) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a1),s(t_h4s_integers_int,V_b2))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a2),s(t_h4s_integers_int,V_b1))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_b1),s(t_h4s_integers_int,V_b2)))))))))).
fof(ah4s_fracs_FRACu_u_SUBu_u_CALCULATE, axiom, ![V_b2, V_b1, V_a2, V_a1]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b1)))) => (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b2)))) => s(t_h4s_fracs_frac,h4s_fracs_fracu_u_sub(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a1),s(t_h4s_integers_int,V_b1))))),s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a2),s(t_h4s_integers_int,V_b2))))))) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_sub(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a1),s(t_h4s_integers_int,V_b2))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a2),s(t_h4s_integers_int,V_b1))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_b1),s(t_h4s_integers_int,V_b2)))))))))).
fof(ah4s_rats_ratu_u_sgnu_u_def, axiom, ![V_r]: s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r))) = s(t_h4s_integers_int,h4s_fracs_fracu_u_sgn(s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,V_r)))))).
fof(ah4s_rats_ratu_u_addu_u_def, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,V_r1))),s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,V_r2)))))))).
fof(ah4s_rats_ratu_u_subu_u_def, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_sub(s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,V_r1))),s(t_h4s_fracs_frac,h4s_rats_repu_u_rat(s(t_h4s_rats_rat,V_r2)))))))).
fof(ah4s_rats_ratu_u_lesu_u_def, axiom, ![V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2)))) <=> s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r2),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_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
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_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_SGNu_u_CONG, axiom, ![V_f1]: s(t_h4s_integers_int,h4s_fracs_fracu_u_sgn(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_fracs_fracu_u_sgn(s(t_h4s_fracs_frac,V_f1)))).
fof(ah4s_rats_RATu_u_ADDu_u_CONGu_c0, axiom, ![V_y, V_x]: s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,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_fracs_frac,V_y))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_x),s(t_h4s_fracs_frac,V_y)))))).
fof(ah4s_rats_RATu_u_ADDu_u_CONGu_c1, axiom, ![V_y, V_x]: s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_x),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_y))))))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_x),s(t_h4s_fracs_frac,V_y)))))).
fof(ah4s_rats_RATu_u_ADDu_u_CALCULATE, axiom, ![V_f2, V_f1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f2))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))))).
fof(ah4s_rats_RATu_u_SUBu_u_CALCULATE, axiom, ![V_f2, V_f1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f2))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_sub(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))))).
fof(ch4s_rats_RATu_u_LESu_u_TRANS, conjecture, ![V_r3, V_r2, V_r1]: ((p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2)))) & p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,V_r3))))) => p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r3)))))).
