%   ORIGINAL: h4/ratRing/rat__ring__thms_c41
% 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/ratRing/rat__r__interp__vl__def: h4/ratRing/rat__r__interp__vl = h4/ringNorm/r__interp__vl (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/ring/ring__accessors_c3: !f1 f0 f a0 a. h4/ring/ring__RM (h4/ring/ring0 a a0 f f0 f1) = f0
% Assm: h4/ratRing/rat__interp__p__def: h4/ratRing/rat__interp__p = h4/ringNorm/interp__p (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ring/ring__accessors_c2: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm: h4/ratRing/rat__r__ivl__aux__def: h4/ratRing/rat__r__ivl__aux = h4/ringNorm/r__ivl__aux (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ringNorm/interp__p__def_c2: !vm r p2 p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Pplus p1 p2) = h4/ring/ring__RP r (h4/ringNorm/interp__p r vm p1) (h4/ringNorm/interp__p r vm p2)
% Assm: h4/ringNorm/interp__p__def_c0: !vm r c. h4/ringNorm/interp__p r vm (h4/ringNorm/Pconst c) = c
% Assm: h4/ringNorm/interp__p__def_c4: !vm r p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Popp p1) = h4/ring/ring__RN r (h4/ringNorm/interp__p r vm p1)
% Assm: h4/ringNorm/interp__p__def_c3: !vm r p2 p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Pmult p1 p2) = h4/ring/ring__RM r (h4/ringNorm/interp__p r vm p1) (h4/ringNorm/interp__p r vm p2)
% Assm: h4/ring/ring__accessors_c4: !f1 f0 f a0 a. h4/ring/ring__RN (h4/ring/ring0 a a0 f f0 f1) = f1
% Assm: h4/ringNorm/interp__p__def_c1: !vm r i. h4/ringNorm/interp__p r vm (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm: h4/ringNorm/ivl__aux__def_c0: !x vm r. h4/ringNorm/r__ivl__aux r vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/ringNorm/ivl__aux__def_c1: !x_27 x vm t_27 r. h4/ringNorm/r__ivl__aux r vm x (h4/list/CONS x_27 t_27) = h4/ring/ring__RM r (h4/quote/varmap__find x vm) (h4/ringNorm/r__ivl__aux r vm x_27 t_27)
% Assm: h4/ratRing/rat__ring__thms_c39: !x vm. h4/ratRing/rat__r__ivl__aux vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/ratRing/rat__ring__thms_c40: !x_27 x vm t_27. h4/ratRing/rat__r__ivl__aux vm x (h4/list/CONS x_27 t_27) = h4/rat/rat__mul (h4/quote/varmap__find x vm) (h4/ratRing/rat__r__ivl__aux vm x_27 t_27)
% Assm: h4/ratRing/rat__ring__thms_c3: !vm i. h4/ratRing/rat__interp__p vm (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm: h4/ratRing/rat__ring__thms_c2: !vm c. h4/ratRing/rat__interp__p vm (h4/ringNorm/Pconst c) = c
% Assm: h4/ratRing/rat__ring__thms_c4: !vm p2 p1. h4/ratRing/rat__interp__p vm (h4/ringNorm/Pplus p1 p2) = h4/rat/rat__add (h4/ratRing/rat__interp__p vm p1) (h4/ratRing/rat__interp__p vm p2)
% Assm: h4/ratRing/rat__ring__thms_c5: !vm p2 p1. h4/ratRing/rat__interp__p vm (h4/ringNorm/Pmult p1 p2) = h4/rat/rat__mul (h4/ratRing/rat__interp__p vm p1) (h4/ratRing/rat__interp__p vm p2)
% Assm: h4/ratRing/rat__ring__thms_c6: !vm p1. h4/ratRing/rat__interp__p vm (h4/ringNorm/Popp p1) = h4/rat/rat__ainv (h4/ratRing/rat__interp__p vm p1)
% Assm: h4/ratRing/rat__ring__thms_c1: !vm p. h4/ratRing/rat__interp__p vm p = h4/ratRing/rat__r__interp__cs vm (h4/ratRing/rat__polynom__simplify p)
% Assm: h4/ratRing/rat__r__interp__cs__def: h4/ratRing/rat__r__interp__cs = h4/ringNorm/r__interp__cs (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__polynom__simplify__def: h4/ratRing/rat__polynom__simplify = h4/ringNorm/polynom__simplify (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ring/ring__accessors_c1: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm: h4/ringNorm/polynom__simplify__ok: !r. h4/ring/is__ring r ==> (!vm p. h4/ringNorm/r__interp__cs r vm (h4/ringNorm/polynom__simplify r p) = h4/ringNorm/interp__p r vm p)
% Assm: h4/ratRing/RAT__IS__RING: h4/ring/is__ring (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__r__varlist__insert__def: h4/ratRing/rat__r__varlist__insert = h4/ringNorm/r__varlist__insert (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__r__monom__insert__def: h4/ratRing/rat__r__monom__insert = h4/ringNorm/r__monom__insert (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__r__canonical__sum__simplify__def: h4/ratRing/rat__r__canonical__sum__simplify = h4/ringNorm/r__canonical__sum__simplify (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__r__interp__m__def: h4/ratRing/rat__r__interp__m = h4/ringNorm/r__interp__m (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__r__ics__aux__def: h4/ratRing/rat__r__ics__aux = h4/ringNorm/r__ics__aux (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__r__interp__sp__def: h4/ratRing/rat__r__interp__sp = h4/ringNorm/r__interp__sp (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__r__canonical__sum__scalar2__def: h4/ratRing/rat__r__canonical__sum__scalar2 = h4/ringNorm/r__canonical__sum__scalar2 (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__ring__thms_c21: !t2 l2 l1 c2. h4/ratRing/rat__r__varlist__insert l1 (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/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (h4/ratRing/rat__r__varlist__insert l1 t2))
% Assm: h4/ratRing/rat__ring__thms_c12: !t2 t1 l2 l1 c1. h4/ratRing/rat__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/ratRing/rat__r__canonical__sum__merge t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/rat/rat__add c1 (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 (h4/ratRing/rat__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__varlist l2 (h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/ratRing/rat__ring__thms_c13: !t2 t1 l2 l1 c2. h4/ratRing/rat__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/ratRing/rat__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) c2) l1 (h4/ratRing/rat__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/ratRing/rat__ring__thms_c14: !t2 t1 l2 l1. h4/ratRing/rat__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/ratRing/rat__r__canonical__sum__merge t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 (h4/ratRing/rat__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__varlist l2 (h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/ratRing/rat__ring__thms_c36: !t l c. h4/ratRing/rat__r__canonical__sum__simplify (h4/canonical/Cons__monom c l t) = h4/bool/COND (c = h4/rat/rat__of__num h4/num/0) (h4/ratRing/rat__r__canonical__sum__simplify t) (h4/bool/COND (c = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/canonical/Cons__varlist l (h4/ratRing/rat__r__canonical__sum__simplify t)) (h4/canonical/Cons__monom c l (h4/ratRing/rat__r__canonical__sum__simplify t)))
% Assm: h4/ratRing/rat__ring__thms_c22: !t2 l2 l1. h4/ratRing/rat__r__varlist__insert l1 (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/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 t2) (h4/canonical/Cons__varlist l2 (h4/ratRing/rat__r__varlist__insert l1 t2))
% Assm: h4/ratRing/rat__ring__thms_c19: !t2 l2 l1 c1. h4/ratRing/rat__r__monom__insert c1 l1 (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/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (h4/rat/rat__add c1 (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 t2) (h4/canonical/Cons__varlist l2 (h4/ratRing/rat__r__monom__insert c1 l1 t2))
% Assm: h4/ratRing/rat__r__canonical__sum__scalar3__def: h4/ratRing/rat__r__canonical__sum__scalar3 = h4/ringNorm/r__canonical__sum__scalar3 (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__r__spolynom__simplify__def: h4/ratRing/rat__r__spolynom__simplify = h4/ringNorm/r__spolynom__simplify (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/rat/RAT__1__NOT__0: ~(h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/rat/rat__of__num h4/num/0)
% Assm: h4/ratRing/rat__polynom__normalize__def: h4/ratRing/rat__polynom__normalize = h4/ringNorm/polynom__normalize (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/rat/RAT__NO__IDDIV: !r2 r1. h4/rat/rat__mul r1 r2 = r2 <=> r1 = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) \/ r2 = h4/rat/rat__of__num h4/num/0
% Assm: h4/rat/RAT__MUL__RINV: !a. ~(a = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__mul a (h4/rat/rat__minv a) = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/ringNorm/varlist__insert__def_c1: !t2 r l2 l1. h4/ringNorm/r__varlist__insert r l1 (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/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (h4/ring/ring__RP r (h4/ring/ring__R1 r) (h4/ring/ring__R1 r)) l1 t2) (h4/canonical/Cons__varlist l2 (h4/ringNorm/r__varlist__insert r l1 t2))
% Assm: h4/ringNorm/varlist__insert__def_c2: !r l1. h4/ringNorm/r__varlist__insert r l1 h4/canonical/Nil__monom = h4/canonical/Cons__varlist l1 h4/canonical/Nil__monom
% Assm: h4/ringNorm/varlist__insert__def_c0: !t2 r l2 l1 c2. h4/ringNorm/r__varlist__insert r l1 (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/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (h4/ring/ring__RP r (h4/ring/ring__R1 r) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (h4/ringNorm/r__varlist__insert r l1 t2))
% Assm: h4/ringNorm/canonical__sum__merge__def_c4: !s1 r. h4/ringNorm/r__canonical__sum__merge r s1 h4/canonical/Nil__monom = s1
% Assm: h4/ringNorm/canonical__sum__merge__def_c2: !t2 t1 r l2 l1 c2. h4/ringNorm/r__canonical__sum__merge r (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/ringNorm/r__canonical__sum__merge r t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/ring/ring__RP r (h4/ring/ring__R1 r) c2) l1 (h4/ringNorm/r__canonical__sum__merge r t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/ringNorm/canonical__sum__merge__def_c0: !t2 t1 r l2 l1 c2 c1. h4/ringNorm/r__canonical__sum__merge r (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/ringNorm/r__canonical__sum__merge r t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/ring/ring__RP r c1 c2) l1 (h4/ringNorm/r__canonical__sum__merge r t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/ringNorm/canonical__sum__merge__def_c6: !v8 v7 r. h4/ringNorm/r__canonical__sum__merge r h4/canonical/Nil__monom (h4/canonical/Cons__varlist v7 v8) = h4/canonical/Cons__varlist v7 v8
% Assm: h4/ringNorm/canonical__sum__merge__def_c5: !v6 v5 v4 r. h4/ringNorm/r__canonical__sum__merge r h4/canonical/Nil__monom (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% Assm: h4/ringNorm/canonical__sum__merge__def_c3: !t2 t1 r l2 l1. h4/ringNorm/r__canonical__sum__merge r (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/ringNorm/r__canonical__sum__merge r t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/ring/ring__RP r (h4/ring/ring__R1 r) (h4/ring/ring__R1 r)) l1 (h4/ringNorm/r__canonical__sum__merge r t1 t2)) (h4/canonical/Cons__varlist l2 (h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/ringNorm/canonical__sum__merge__def_c1: !t2 t1 r l2 l1 c1. h4/ringNorm/r__canonical__sum__merge r (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/ringNorm/r__canonical__sum__merge r t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/ring/ring__RP r c1 (h4/ring/ring__R1 r)) l1 (h4/ringNorm/r__canonical__sum__merge r t1 t2)) (h4/canonical/Cons__varlist l2 (h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/ratRing/rat__r__canonical__sum__merge__def: h4/ratRing/rat__r__canonical__sum__merge = h4/ringNorm/r__canonical__sum__merge (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ringNorm/canonical__sum__simplify__def_c2: !r. h4/ringNorm/r__canonical__sum__simplify r h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/ringNorm/canonical__sum__simplify__def_c1: !t r l. h4/ringNorm/r__canonical__sum__simplify r (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__varlist l (h4/ringNorm/r__canonical__sum__simplify r t)
% Assm: h4/ringNorm/canonical__sum__simplify__def_c0: !t r l c. h4/ringNorm/r__canonical__sum__simplify r (h4/canonical/Cons__monom c l t) = h4/bool/COND (c = h4/ring/ring__R0 r) (h4/ringNorm/r__canonical__sum__simplify r t) (h4/bool/COND (c = h4/ring/ring__R1 r) (h4/canonical/Cons__varlist l (h4/ringNorm/r__canonical__sum__simplify r t)) (h4/canonical/Cons__monom c l (h4/ringNorm/r__canonical__sum__simplify r t)))
% Assm: h4/ring/ring__accessors_c0: !f1 f0 f a0 a. h4/ring/ring__R0 (h4/ring/ring0 a a0 f f0 f1) = a
% Assm: h4/ringNorm/monom__insert__def_c2: !r l1 c1. h4/ringNorm/r__monom__insert r c1 l1 h4/canonical/Nil__monom = h4/canonical/Cons__monom c1 l1 h4/canonical/Nil__monom
% Assm: h4/ringNorm/monom__insert__def_c0: !t2 r l2 l1 c2 c1. h4/ringNorm/r__monom__insert r c1 l1 (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/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (h4/ring/ring__RP r c1 c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (h4/ringNorm/r__monom__insert r c1 l1 t2))
% Assm: h4/ringNorm/monom__insert__def_c1: !t2 r l2 l1 c1. h4/ringNorm/r__monom__insert r c1 l1 (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/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (h4/ring/ring__RP r c1 (h4/ring/ring__R1 r)) l1 t2) (h4/canonical/Cons__varlist l2 (h4/ringNorm/r__monom__insert r c1 l1 t2))
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/integer/INT__MUL__LZERO: !x. h4/integer/int__mul (h4/integer/int__of__num h4/num/0) x = h4/integer/int__of__num h4/num/0
% Assm: h4/rat/RAT__ABS__EQUIV: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat f2 <=> h4/rat/rat__equiv f1 f2
% Assm: h4/integer/INT__INJ: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm: h4/integer/INT__MUL__LID: !x. h4/integer/int__mul (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm: h4/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/frac__1__def: h4/frac/frac__1 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/rat/rat__00: h4/rat/rat__of__num h4/num/0 = h4/rat/abs__rat h4/frac/frac__0
% Assm: h4/rat/rat__equiv__def: !f2 f1. h4/rat/rat__equiv f1 f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm: h4/integer/INT__LT__REDUCE_c0: !n. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) <=> T
% Assm: h4/rat/rat__10: h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/rat/abs__rat h4/frac/frac__1
% Assm: h4/frac/frac__0__def: h4/frac/frac__0 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/frac/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/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/rat/RAT__MUL__RID: !a. h4/rat/rat__mul a (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = a
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/rat/RAT__EQ__RMUL: !r3 r2 r1. ~(r3 = h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__mul r1 r3 = h4/rat/rat__mul r2 r3 <=> r1 = r2)
% Assm: h4/rat/RAT__MUL__RZERO: !r1. h4/rat/rat__mul r1 (h4/rat/rat__of__num h4/num/0) = h4/rat/rat__of__num h4/num/0
% Assm: h4/rat/RAT__MUL__ASSOC: !c b a. h4/rat/rat__mul a (h4/rat/rat__mul b c) = h4/rat/rat__mul (h4/rat/rat__mul a b) c
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% 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__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/rat/RAT__MUL__LID: !a. h4/rat/rat__mul (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) a = a
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> 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/numRing/num__rewrites_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numRing/num__rewrites_c75: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> T
% 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/intExtension/SGN__def: !x. h4/intExtension/SGN x = h4/bool/COND (x = h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num h4/num/0) (h4/bool/COND (h4/integer/int__lt x (h4/integer/int__of__num h4/num/0)) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/integerRing/int__rewrites_c14: h4/integer/int__0 = h4/integer/int__of__num h4/num/0
% 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__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/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/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__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_c52: !c. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_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_c45: !vm a. h4/integerRing/int__r__ics__aux vm a h4/canonical/Nil__monom = a
% Assm: h4/numRing/num__rewrites_c37: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% 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_c50: !vm t l c. h4/integerRing/int__r__interp__cs vm (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__m vm c l) t
% Assm: h4/integerRing/int__ring__thms_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_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_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
% Goal: !vm. h4/ratRing/rat__r__interp__vl vm h4/list/NIL = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
%   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_ratRings_ratu_u_ru_u_interpu_u_vlu_u_def]: h4/ratRing/rat__r__interp__vl = h4/ringNorm/r__interp__vl (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_rings_ringu_u_accessorsu_c3]: !f1 f0 f a0 a. h4/ring/ring__RM (h4/ring/ring0 a a0 f f0 f1) = f0
% Assm [h4s_ratRings_ratu_u_interpu_u_pu_u_def]: h4/ratRing/rat__interp__p = h4/ringNorm/interp__p (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_rings_ringu_u_accessorsu_c2]: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm [h4s_ratRings_ratu_u_ru_u_ivlu_u_auxu_u_def]: h4/ratRing/rat__r__ivl__aux = h4/ringNorm/r__ivl__aux (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c2]: !vm r p2 p1. happ (happ (h4/ringNorm/interp__p r) vm) (h4/ringNorm/Pplus p1 p2) = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ringNorm/interp__p r) vm) p1)) (happ (happ (h4/ringNorm/interp__p r) vm) p2)
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c0]: !vm r c. happ (happ (h4/ringNorm/interp__p r) vm) (h4/ringNorm/Pconst c) = c
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c4]: !vm r p1. happ (happ (h4/ringNorm/interp__p r) vm) (h4/ringNorm/Popp p1) = happ (h4/ring/ring__RN r) (happ (happ (h4/ringNorm/interp__p r) vm) p1)
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c3]: !vm r p2 p1. happ (happ (h4/ringNorm/interp__p r) vm) (h4/ringNorm/Pmult p1 p2) = happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ringNorm/interp__p r) vm) p1)) (happ (happ (h4/ringNorm/interp__p r) vm) p2)
% Assm [h4s_rings_ringu_u_accessorsu_c4]: !f1 f0 f a0 a. h4/ring/ring__RN (h4/ring/ring0 a a0 f f0 f1) = f1
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c1]: !vm r i. happ (happ (h4/ringNorm/interp__p r) vm) (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm [h4s_ringNorms_ivlu_u_auxu_u_defu_c0]: !x vm r. happ (happ (happ (h4/ringNorm/r__ivl__aux r) vm) x) h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_ringNorms_ivlu_u_auxu_u_defu_c1]: !x_27 x vm t_27 r. happ (happ (happ (h4/ringNorm/r__ivl__aux r) vm) x) (h4/list/CONS x_27 t_27) = happ (happ (h4/ring/ring__RM r) (h4/quote/varmap__find x vm)) (happ (happ (happ (h4/ringNorm/r__ivl__aux r) vm) x_27) t_27)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c39]: !x vm. happ (happ (happ h4/ratRing/rat__r__ivl__aux vm) x) h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c40]: !x_27 x vm t_27. happ (happ (happ h4/ratRing/rat__r__ivl__aux vm) x) (h4/list/CONS x_27 t_27) = happ (happ h4/rat/rat__mul (h4/quote/varmap__find x vm)) (happ (happ (happ h4/ratRing/rat__r__ivl__aux vm) x_27) t_27)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c3]: !vm i. happ (happ h4/ratRing/rat__interp__p vm) (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c2]: !vm c. happ (happ h4/ratRing/rat__interp__p vm) (h4/ringNorm/Pconst c) = c
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c4]: !vm p2 p1. happ (happ h4/ratRing/rat__interp__p vm) (h4/ringNorm/Pplus p1 p2) = happ (happ h4/rat/rat__add (happ (happ h4/ratRing/rat__interp__p vm) p1)) (happ (happ h4/ratRing/rat__interp__p vm) p2)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c5]: !vm p2 p1. happ (happ h4/ratRing/rat__interp__p vm) (h4/ringNorm/Pmult p1 p2) = happ (happ h4/rat/rat__mul (happ (happ h4/ratRing/rat__interp__p vm) p1)) (happ (happ h4/ratRing/rat__interp__p vm) p2)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c6]: !vm p1. happ (happ h4/ratRing/rat__interp__p vm) (h4/ringNorm/Popp p1) = happ h4/rat/rat__ainv (happ (happ h4/ratRing/rat__interp__p vm) p1)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c1]: !vm p. happ (happ h4/ratRing/rat__interp__p vm) p = happ (happ h4/ratRing/rat__r__interp__cs vm) (happ h4/ratRing/rat__polynom__simplify p)
% Assm [h4s_ratRings_ratu_u_ru_u_interpu_u_csu_u_def]: h4/ratRing/rat__r__interp__cs = h4/ringNorm/r__interp__cs (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_polynomu_u_simplifyu_u_def]: h4/ratRing/rat__polynom__simplify = h4/ringNorm/polynom__simplify (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_rings_ringu_u_accessorsu_c1]: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm [h4s_ringNorms_polynomu_u_simplifyu_u_ok]: !r. h4/ring/is__ring r ==> (!vm p. happ (happ (h4/ringNorm/r__interp__cs r) vm) (happ (h4/ringNorm/polynom__simplify r) p) = happ (happ (h4/ringNorm/interp__p r) vm) p)
% Assm [h4s_ratRings_RATu_u_ISu_u_RING]: h4/ring/is__ring (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ru_u_varlistu_u_insertu_u_def]: h4/ratRing/rat__r__varlist__insert = h4/ringNorm/r__varlist__insert (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ru_u_monomu_u_insertu_u_def]: h4/ratRing/rat__r__monom__insert = h4/ringNorm/r__monom__insert (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_simplifyu_u_def]: h4/ratRing/rat__r__canonical__sum__simplify = h4/ringNorm/r__canonical__sum__simplify (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ru_u_interpu_u_mu_u_def]: h4/ratRing/rat__r__interp__m = h4/ringNorm/r__interp__m (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ru_u_icsu_u_auxu_u_def]: h4/ratRing/rat__r__ics__aux = h4/ringNorm/r__ics__aux (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ru_u_interpu_u_spu_u_def]: h4/ratRing/rat__r__interp__sp = h4/ringNorm/r__interp__sp (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar2u_u_def]: h4/ratRing/rat__r__canonical__sum__scalar2 = h4/ringNorm/r__canonical__sum__scalar2 (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c21]: !t2 l2 l1 c2. happ (happ h4/ratRing/rat__r__varlist__insert l1) (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/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (happ (happ h4/ratRing/rat__r__varlist__insert l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c12]: !t2 t1 l2 l1 c1. happ (happ h4/ratRing/rat__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 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add c1) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) t2)) (h4/canonical/Cons__varlist l2 (happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c13]: !t2 t1 l2 l1 c2. happ (happ h4/ratRing/rat__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 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) c2) l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) t2)) (h4/canonical/Cons__monom c2 l2 (happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c14]: !t2 t1 l2 l1. happ (happ h4/ratRing/rat__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 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) t2)) (h4/canonical/Cons__varlist l2 (happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c36]: !t l c. ?v. (v <=> c = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ (?v'. (v' <=> c = h4/rat/rat__of__num h4/num/0) /\ happ h4/ratRing/rat__r__canonical__sum__simplify (h4/canonical/Cons__monom c l t) = h4/bool/COND v' (happ h4/ratRing/rat__r__canonical__sum__simplify t) (h4/bool/COND v (h4/canonical/Cons__varlist l (happ h4/ratRing/rat__r__canonical__sum__simplify t)) (h4/canonical/Cons__monom c l (happ h4/ratRing/rat__r__canonical__sum__simplify t))))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c22]: !t2 l2 l1. happ (happ h4/ratRing/rat__r__varlist__insert l1) (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/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 t2) (h4/canonical/Cons__varlist l2 (happ (happ h4/ratRing/rat__r__varlist__insert l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c19]: !t2 l2 l1 c1. happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) (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/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add c1) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 t2) (h4/canonical/Cons__varlist l2 (happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) t2))
% Assm [h4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar3u_u_def]: h4/ratRing/rat__r__canonical__sum__scalar3 = h4/ringNorm/r__canonical__sum__scalar3 (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_ru_u_spolynomu_u_simplifyu_u_def]: h4/ratRing/rat__r__spolynom__simplify = h4/ringNorm/r__spolynom__simplify (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_rats_RATu_u_1u_u_NOTu_u_0]: ~(h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/rat/rat__of__num h4/num/0)
% Assm [h4s_ratRings_ratu_u_polynomu_u_normalizeu_u_def]: h4/ratRing/rat__polynom__normalize = h4/ringNorm/polynom__normalize (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_rats_RATu_u_NOu_u_IDDIV]: !r2 r1. happ (happ h4/rat/rat__mul r1) r2 = r2 <=> r1 = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) \/ r2 = h4/rat/rat__of__num h4/num/0
% Assm [h4s_rats_RATu_u_MULu_u_RINV]: !a. ~(a = h4/rat/rat__of__num h4/num/0) ==> happ (happ h4/rat/rat__mul a) (h4/rat/rat__minv a) = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_ringNorms_varlistu_u_insertu_u_defu_c1]: !t2 r l2 l1. happ (happ (h4/ringNorm/r__varlist__insert r) l1) (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/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) (h4/ring/ring__R1 r)) l1 t2) (h4/canonical/Cons__varlist l2 (happ (happ (h4/ringNorm/r__varlist__insert r) l1) t2))
% Assm [h4s_ringNorms_varlistu_u_insertu_u_defu_c2]: !r l1. happ (happ (h4/ringNorm/r__varlist__insert r) l1) h4/canonical/Nil__monom = h4/canonical/Cons__varlist l1 h4/canonical/Nil__monom
% Assm [h4s_ringNorms_varlistu_u_insertu_u_defu_c0]: !t2 r l2 l1 c2. happ (happ (h4/ringNorm/r__varlist__insert r) l1) (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/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (happ (happ (h4/ringNorm/r__varlist__insert r) l1) t2))
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c4]: !s1 r. happ (happ (h4/ringNorm/r__canonical__sum__merge r) s1) h4/canonical/Nil__monom = s1
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c2]: !t2 t1 r l2 l1 c2. happ (happ (h4/ringNorm/r__canonical__sum__merge r) (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 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) c2) l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) t2)) (h4/canonical/Cons__monom c2 l2 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__varlist l1 t1)) t2))
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c0]: !t2 t1 r l2 l1 c2 c1. happ (happ (h4/ringNorm/r__canonical__sum__merge r) (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 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) c2) l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) t2)) (h4/canonical/Cons__monom c2 l2 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__monom c1 l1 t1)) t2))
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c6]: !v8 v7 r. happ (happ (h4/ringNorm/r__canonical__sum__merge r) h4/canonical/Nil__monom) (h4/canonical/Cons__varlist v7 v8) = h4/canonical/Cons__varlist v7 v8
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c5]: !v6 v5 v4 r. happ (happ (h4/ringNorm/r__canonical__sum__merge r) h4/canonical/Nil__monom) (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c3]: !t2 t1 r l2 l1. happ (happ (h4/ringNorm/r__canonical__sum__merge r) (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 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) (h4/ring/ring__R1 r)) l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) t2)) (h4/canonical/Cons__varlist l2 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__varlist l1 t1)) t2))
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c1]: !t2 t1 r l2 l1 c1. happ (happ (h4/ringNorm/r__canonical__sum__merge r) (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 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) (h4/ring/ring__R1 r)) l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) t2)) (h4/canonical/Cons__varlist l2 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__monom c1 l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_mergeu_u_def]: h4/ratRing/rat__r__canonical__sum__merge = h4/ringNorm/r__canonical__sum__merge (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_simplifyu_u_defu_c2]: !r. happ (h4/ringNorm/r__canonical__sum__simplify r) h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_simplifyu_u_defu_c1]: !t r l. happ (h4/ringNorm/r__canonical__sum__simplify r) (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__varlist l (happ (h4/ringNorm/r__canonical__sum__simplify r) t)
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_simplifyu_u_defu_c0]: !t r l c. ?v. (v <=> c = h4/ring/ring__R1 r) /\ (?v'. (v' <=> c = h4/ring/ring__R0 r) /\ happ (h4/ringNorm/r__canonical__sum__simplify r) (h4/canonical/Cons__monom c l t) = h4/bool/COND v' (happ (h4/ringNorm/r__canonical__sum__simplify r) t) (h4/bool/COND v (h4/canonical/Cons__varlist l (happ (h4/ringNorm/r__canonical__sum__simplify r) t)) (h4/canonical/Cons__monom c l (happ (h4/ringNorm/r__canonical__sum__simplify r) t))))
% Assm [h4s_rings_ringu_u_accessorsu_c0]: !f1 f0 f a0 a. h4/ring/ring__R0 (h4/ring/ring0 a a0 f f0 f1) = a
% Assm [h4s_ringNorms_monomu_u_insertu_u_defu_c2]: !r l1 c1. happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) h4/canonical/Nil__monom = h4/canonical/Cons__monom c1 l1 h4/canonical/Nil__monom
% Assm [h4s_ringNorms_monomu_u_insertu_u_defu_c0]: !t2 r l2 l1 c2 c1. happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) (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/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) t2))
% Assm [h4s_ringNorms_monomu_u_insertu_u_defu_c1]: !t2 r l2 l1 c1. happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) (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/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) (h4/ring/ring__R1 r)) l1 t2) (h4/canonical/Cons__varlist l2 (happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) t2))
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_integers_INTu_u_MULu_u_LZERO]: !x. h4/integer/int__mul (h4/integer/int__of__num h4/num/0) x = h4/integer/int__of__num h4/num/0
% Assm [h4s_rats_RATu_u_ABSu_u_EQUIV]: !f2 f1. h4/rat/abs__rat f1 = h4/rat/abs__rat f2 <=> h4/rat/rat__equiv f1 f2
% Assm [h4s_integers_INTu_u_INJ]: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm [h4s_integers_INTu_u_MULu_u_LID]: !x. h4/integer/int__mul (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm [h4s_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_fracu_u_1u_u_def]: h4/frac/frac__1 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_rats_ratu_u_00]: h4/rat/rat__of__num h4/num/0 = h4/rat/abs__rat h4/frac/frac__0
% Assm [h4s_rats_ratu_u_equivu_u_def]: !f2 f1. h4/rat/rat__equiv f1 f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm [h4s_integers_INTu_u_LTu_u_REDUCEu_c0]: !n. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) <=> T
% Assm [h4s_rats_ratu_u_10]: h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/rat/abs__rat h4/frac/frac__1
% Assm [h4s_fracs_fracu_u_0u_u_def]: h4/frac/frac__0 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__of__num h4/num/0) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_fracs_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_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_rats_RATu_u_MULu_u_RID]: !a. happ (happ h4/rat/rat__mul a) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = a
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_rats_RATu_u_EQu_u_RMUL]: !r3 r2 r1. ~(r3 = h4/rat/rat__of__num h4/num/0) ==> (happ (happ h4/rat/rat__mul r1) r3 = happ (happ h4/rat/rat__mul r2) r3 <=> r1 = r2)
% Assm [h4s_rats_RATu_u_MULu_u_RZERO]: !r1. happ (happ h4/rat/rat__mul r1) (h4/rat/rat__of__num h4/num/0) = h4/rat/rat__of__num h4/num/0
% Assm [h4s_rats_RATu_u_MULu_u_ASSOC]: !c b a. happ (happ h4/rat/rat__mul a) (happ (happ h4/rat/rat__mul b) c) = happ (happ h4/rat/rat__mul (happ (happ h4/rat/rat__mul a) b)) c
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% 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_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_rats_RATu_u_MULu_u_LID]: !a. happ (happ h4/rat/rat__mul (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) a = a
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> 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_numRings_numu_u_rewritesu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numRings_numu_u_rewritesu_c75]: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> T
% 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_intExtensions_SGNu_u_def]: !x. ?v. (v <=> x = h4/integer/int__of__num h4/num/0) /\ h4/intExtension/SGN x = h4/bool/COND v (h4/integer/int__of__num h4/num/0) (h4/bool/COND (h4/integer/int__lt x (h4/integer/int__of__num h4/num/0)) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_integerRings_intu_u_rewritesu_c14]: h4/integer/int__0 = h4/integer/int__of__num h4/num/0
% 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_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_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_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_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_c52]: !c. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_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_c45]: !vm a. h4/integerRing/int__r__ics__aux vm a h4/canonical/Nil__monom = a
% Assm [h4s_numRings_numu_u_rewritesu_c37]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% 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_c50]: !vm t l c. h4/integerRing/int__r__interp__cs vm (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__m vm c l) t
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_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_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_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
% Goal: !vm. happ (happ h4/ratRing/rat__r__interp__vl vm) h4/list/NIL = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
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_Q1331050,TV_Q1331046]: ![V_f, V_g]: (![V_x]: s(TV_Q1331046,happ(s(t_fun(TV_Q1331050,TV_Q1331046),V_f),s(TV_Q1331050,V_x))) = s(TV_Q1331046,happ(s(t_fun(TV_Q1331050,TV_Q1331046),V_g),s(TV_Q1331050,V_x))) => s(t_fun(TV_Q1331050,TV_Q1331046),V_f) = s(t_fun(TV_Q1331050,TV_Q1331046),V_g))).
fof(ah4s_ratRings_ratu_u_ru_u_interpu_u_vlu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_interpu_u_vl) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat)),h4s_ringnorms_ru_u_interpu_u_vl(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
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_rings_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_ratRings_ratu_u_interpu_u_pu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_rings_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_ratRings_ratu_u_ru_u_ivlu_u_auxu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_ivlu_u_aux) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c2, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_pplus(s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2)))))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_r, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_pconst(s(TV_u_27a,V_c))))) = s(TV_u_27a,V_c)).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c4, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_popp(s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1)))))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c3, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_pmult(s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2)))))).
fof(ah4s_rings_ringu_u_accessorsu_c4, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,TV_u_27a),V_f1)).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_r, V_i]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_pvar(s(t_h4s_quotes_index,V_i))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_ringNorms_ivlu_u_auxu_u_defu_c0, axiom, ![TV_u_27a]: ![V_x, V_vm, V_r]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_ringNorms_ivlu_u_auxu_u_defu_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_vm, V_tu_27, V_r]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),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(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_quotes_index,V_xu_27))),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c39, axiom, ![V_x, V_vm]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_ivlu_u_aux),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(t_h4s_rats_rat,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm)))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c40, axiom, ![V_xu_27, V_x, V_vm, V_tu_27]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_ivlu_u_aux),s(t_h4s_quotes_varmap(t_h4s_rats_rat),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_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_ivlu_u_aux),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_quotes_index,V_xu_27))),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c3, axiom, ![V_vm, V_i]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),h4s_ringnorms_pvar(s(t_h4s_quotes_index,V_i))))) = s(t_h4s_rats_rat,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm)))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c2, axiom, ![V_vm, V_c]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),h4s_ringnorms_pconst(s(t_h4s_rats_rat,V_c))))) = s(t_h4s_rats_rat,V_c)).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c4, axiom, ![V_vm, V_p2, V_p1]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),h4s_ringnorms_pplus(s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p1),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p2))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p1))))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p2)))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c5, axiom, ![V_vm, V_p2, V_p1]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),h4s_ringnorms_pmult(s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p1),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p2))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p1))))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p2)))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c6, axiom, ![V_vm, V_p1]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),h4s_ringnorms_popp(s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p1))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p1)))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c1, axiom, ![V_vm, V_p]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_interpu_u_cs),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_polynomu_u_simplify),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p)))))).
fof(ah4s_ratRings_ratu_u_ru_u_interpu_u_csu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_interpu_u_cs) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ringnorms_ru_u_interpu_u_cs(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_polynomu_u_simplifyu_u_def, axiom, s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_polynomu_u_simplify) = s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ringnorms_polynomu_u_simplify(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_rings_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a0)).
fof(ah4s_ringNorms_polynomu_u_simplifyu_u_ok, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_vm, V_p]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_cs(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))).
fof(ah4s_ratRings_RATu_u_ISu_u_RING, axiom, p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv))))))).
fof(ah4s_ratRings_ratu_u_ru_u_varlistu_u_insertu_u_def, axiom, s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_varlistu_u_insert) = s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_ru_u_monomu_u_insertu_u_def, axiom, s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_monomu_u_insert) = s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_simplifyu_u_def, axiom, s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_simplify) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_ru_u_interpu_u_mu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_interpu_u_m) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat))),h4s_ringnorms_ru_u_interpu_u_m(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_ru_u_icsu_u_auxu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_rats_rat,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_icsu_u_aux) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_rats_rat,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat))),h4s_ringnorms_ru_u_icsu_u_aux(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_ru_u_interpu_u_spu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_canonicals_spolynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_interpu_u_sp) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_canonicals_spolynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ringnorms_ru_u_interpu_u_sp(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar2u_u_def, axiom, s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar2) = s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c21, axiom, ![V_t2, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_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_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),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_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_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_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,V_c1))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c14, axiom, ![V_t2, V_t1, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c36, axiom, ![V_t, V_l, V_c]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_rats_rat,V_c) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_rats_rat,V_c) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_simplify),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_bools_cond(s(t_bool,V_vi_),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_simplify),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_simplify),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_simplify),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t)))))))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c22, axiom, ![V_t2, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_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_rats_rat),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_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),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_rats_rat),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_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_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_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c19, axiom, ![V_t2, V_l2, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_monomu_u_insert),s(t_h4s_rats_rat,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,V_c1))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),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_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_monomu_u_insert),s(t_h4s_rats_rat,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar3u_u_def, axiom, s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar3) = s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ratRings_ratu_u_ru_u_spolynomu_u_simplifyu_u_def, axiom, s(t_fun(t_h4s_canonicals_spolynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_spolynomu_u_simplify) = s(t_fun(t_h4s_canonicals_spolynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ringnorms_ru_u_spolynomu_u_simplify(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_rats_RATu_u_1u_u_NOTu_u_0, axiom, ~ (s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_ratRings_ratu_u_polynomu_u_normalizeu_u_def, axiom, s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_polynomu_u_normalize) = s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_rats_RATu_u_NOu_u_IDDIV, axiom, ![V_r2, V_r1]: (s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,V_r2) <=> (s(t_h4s_rats_rat,V_r1) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) | s(t_h4s_rats_rat,V_r2) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_rats_RATu_u_MULu_u_RINV, axiom, ![V_a]: (~ (s(t_h4s_rats_rat,V_a) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,V_a))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_ringNorms_varlistu_u_insertu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t2, V_r, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_varlistu_u_insertu_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_ringNorms_varlistu_u_insertu_u_defu_c0, axiom, ![TV_u_27a]: ![V_t2, V_r, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c4, axiom, ![TV_u_27a]: ![V_s1, V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_s1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_s1)).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_r, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_r, V_l2, V_l1, V_c2, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(TV_u_27a,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c6, axiom, ![TV_u_27a]: ![V_v8, V_v7, V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_v7),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_v8))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_v7),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_v8)))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c5, axiom, ![TV_u_27a]: ![V_v6, V_v5, V_v4, V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_v4),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_v6))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_v4),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_v6)))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c3, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_r, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_r, V_l2, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_mergeu_u_def, axiom, s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_simplifyu_u_defu_c2, axiom, ![TV_u_27a]: ![V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_simplifyu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_r, V_l]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_simplifyu_u_defu_c0, axiom, ![TV_u_27a]: ![V_t, V_r, V_l, V_c]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_c) = s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r)))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(TV_u_27a,V_c) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r)))) & s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_bools_cond(s(t_bool,V_vi_),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))))))))).
fof(ah4s_rings_ringu_u_accessorsu_c0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a)).
fof(ah4s_ringNorms_monomu_u_insertu_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_ringNorms_monomu_u_insertu_u_defu_c0, axiom, ![TV_u_27a]: ![V_t2, V_r, V_l2, V_l1, V_c2, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(TV_u_27a,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_monomu_u_insertu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t2, V_r, V_l2, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_integers_INTu_u_MULu_u_LZERO, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_rats_RATu_u_ABSu_u_EQUIV, axiom, ![V_f2, V_f1]: (s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f1))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,V_f2))) <=> p(s(t_bool,h4s_rats_ratu_u_equiv(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))))).
fof(ah4s_integers_INTu_u_INJ, axiom, ![V_n, V_m]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_integers_INTu_u_MULu_u_LID, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,V_x)).
fof(ah4s_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_fracu_u_1u_u_def, axiom, s(t_h4s_fracs_frac,h4s_fracs_fracu_u_1) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
fof(ah4s_rats_ratu_u_00, axiom, s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_0)))).
fof(ah4s_rats_ratu_u_equivu_u_def, axiom, ![V_f2, V_f1]: (p(s(t_bool,h4s_rats_ratu_u_equiv(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))) <=> s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))))))).
fof(ah4s_integers_INTu_u_LTu_u_REDUCEu_c0, axiom, ![V_n]: s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))))) = s(t_bool,t)).
fof(ah4s_rats_ratu_u_10, axiom, s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_rats_rat,h4s_rats_absu_u_rat(s(t_h4s_fracs_frac,h4s_fracs_fracu_u_1)))).
fof(ah4s_fracs_fracu_u_0u_u_def, axiom, s(t_h4s_fracs_frac,h4s_fracs_fracu_u_0) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
fof(ah4s_fracs_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_numerals_numeralu_u_equ_c1, 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_numerals_numeralu_u_distribu_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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_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_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_rats_RATu_u_MULu_u_RID, axiom, ![V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) = s(t_h4s_rats_rat,V_a)).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_rats_RATu_u_EQu_u_RMUL, axiom, ![V_r3, V_r2, V_r1]: (~ (s(t_h4s_rats_rat,V_r3) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => (s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,V_r3))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r2))),s(t_h4s_rats_rat,V_r3))) <=> s(t_h4s_rats_rat,V_r1) = s(t_h4s_rats_rat,V_r2)))).
fof(ah4s_rats_RATu_u_MULu_u_RZERO, axiom, ![V_r1]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_rats_RATu_u_MULu_u_ASSOC, axiom, ![V_c, V_b, V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_b))),s(t_h4s_rats_rat,V_c))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,V_b))))),s(t_h4s_rats_rat,V_c)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_rats_RATu_u_MULu_u_LID, axiom, ![V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,V_a))) = s(t_h4s_rats_rat,V_a)).
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_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_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_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_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_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_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_intExtensions_SGNu_u_def, axiom, ![V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))))).
fof(ah4s_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_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_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_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_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_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_c52, axiom, ![V_c]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pconst(s(t_h4s_integers_int,V_c))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_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_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_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_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_c50, axiom, ![V_vm, V_t, V_l, V_c]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_cs(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_monom(s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_icsu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_m(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_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_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_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(ch4s_ratRings_ratu_u_ringu_u_thmsu_c41, conjecture, ![V_vm]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_interpu_u_vl),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
