%   ORIGINAL: h4/ratRing/rat__ring__thms_c30
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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__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/ring/ring__accessors_c1: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% 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/ringNorm/canonical__sum__scalar2__def_c0: !t r l0 l c. h4/ringNorm/r__canonical__sum__scalar2 r l0 (h4/canonical/Cons__monom c l t) = h4/ringNorm/r__monom__insert r c (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/ringNorm/r__canonical__sum__scalar2 r l0 t)
% Assm: h4/ringNorm/canonical__sum__scalar2__def_c1: !t r l0 l. h4/ringNorm/r__canonical__sum__scalar2 r l0 (h4/canonical/Cons__varlist l t) = h4/ringNorm/r__varlist__insert r (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/ringNorm/r__canonical__sum__scalar2 r l0 t)
% Assm: h4/ratRing/rat__ring__thms_c27: !t l0 l c. h4/ratRing/rat__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__monom c l t) = h4/ratRing/rat__r__monom__insert c (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/ratRing/rat__r__canonical__sum__scalar2 l0 t)
% Assm: h4/canonical/canonical__sum__scalar2__def_c0: !t sr l0 l c. h4/canonical/canonical__sum__scalar2 sr l0 (h4/canonical/Cons__monom c l t) = h4/canonical/monom__insert sr c (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/canonical/canonical__sum__scalar2 sr l0 t)
% Assm: h4/canonical/canonical__sum__scalar2__def_c1: !t sr l0 l. h4/canonical/canonical__sum__scalar2 sr l0 (h4/canonical/Cons__varlist l t) = h4/canonical/varlist__insert sr (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/canonical/canonical__sum__scalar2 sr l0 t)
% Assm: h4/canonical/canonical__sum__scalar3__def_c0: !t sr l0 l c0 c. h4/canonical/canonical__sum__scalar3 sr c0 l0 (h4/canonical/Cons__monom c l t) = h4/canonical/monom__insert sr (h4/semi__ring/semi__ring__SRM sr c0 c) (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/canonical/canonical__sum__scalar3 sr c0 l0 t)
% Assm: h4/canonical/canonical__sum__scalar3__def_c1: !t sr l0 l c0. h4/canonical/canonical__sum__scalar3 sr c0 l0 (h4/canonical/Cons__varlist l t) = h4/canonical/monom__insert sr c0 (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/canonical/canonical__sum__scalar3 sr c0 l0 t)
% Assm: h4/ringNorm/canonical__sum__scalar2__def_c2: !r l0. h4/ringNorm/r__canonical__sum__scalar2 r l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% 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__ring__thms_c18: !t2 l2 l1 c2 c1. h4/ratRing/rat__r__monom__insert 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/rat/rat__add c1 c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (h4/ratRing/rat__r__monom__insert c1 l1 t2))
% Assm: h4/ringNorm/canonical__sum__scalar3__def_c0: !t r l0 l c0 c. h4/ringNorm/r__canonical__sum__scalar3 r c0 l0 (h4/canonical/Cons__monom c l t) = h4/ringNorm/r__monom__insert r (h4/ring/ring__RM r c0 c) (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/ringNorm/r__canonical__sum__scalar3 r c0 l0 t)
% Assm: h4/ringNorm/canonical__sum__scalar3__def_c1: !t r l0 l c0. h4/ringNorm/r__canonical__sum__scalar3 r c0 l0 (h4/canonical/Cons__varlist l t) = h4/ringNorm/r__monom__insert r c0 (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/ringNorm/r__canonical__sum__scalar3 r c0 l0 t)
% 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/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_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_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/ring/ring__accessors_c3: !f1 f0 f a0 a. h4/ring/ring__RM (h4/ring/ring0 a a0 f f0 f1) = f0
% Assm: h4/canonical/canonical__sum__scalar2__def_c2: !sr l0. h4/canonical/canonical__sum__scalar2 sr l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/numRing/num__monom__insert__def: h4/numRing/num__monom__insert = h4/canonical/monom__insert (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/integerRing/int__r__monom__insert__def: h4/integerRing/int__r__monom__insert = h4/ringNorm/r__monom__insert (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/ringNorm/r__monom__insert__def: !r. h4/ringNorm/r__monom__insert r = h4/canonical/monom__insert (h4/ring/semi__ring__of r)
% Assm: h4/canonical/canonical__sum__scalar3__def_c2: !sr l0 c0. h4/canonical/canonical__sum__scalar3 sr c0 l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/semi__ring/semi__ring__accessors_c3: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm: h4/ratRing/rat__ring__thms_c28: !t l0 l. h4/ratRing/rat__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/ratRing/rat__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/ratRing/rat__r__canonical__sum__scalar2 l0 t)
% 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/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/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__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_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_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/ratRing/rat__ring__thms_c24: !t l c0 c. h4/ratRing/rat__r__canonical__sum__scalar c0 (h4/canonical/Cons__monom c l t) = h4/canonical/Cons__monom (h4/rat/rat__mul c0 c) l (h4/ratRing/rat__r__canonical__sum__scalar c0 t)
% 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__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_c11: !t2 t1 l2 l1 c2 c1. h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/ratRing/rat__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/rat/rat__add c1 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__monom c1 l1 t1) t2))
% Assm: h4/ratRing/rat__ring__thms_c16: !v6 v5 v4. h4/ratRing/rat__r__canonical__sum__merge h4/canonical/Nil__monom (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% 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/numRing/num__ring__thms_c26: !t l0 l c. h4/numRing/num__canonical__sum__scalar2 l0 (h4/canonical/Cons__monom c l t) = h4/numRing/num__monom__insert c (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/numRing/num__canonical__sum__scalar2 l0 t)
% Assm: h4/integerRing/int__ring__thms_c27: !t l0 l c. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__monom__insert c (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar2 l0 t)
% Assm: h4/integerRing/int__ring__thms_c30: !t l0 l c0 c. h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__monom c l t) = h4/integerRing/int__r__monom__insert (h4/integer/int__mul c0 c) (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 t)
% Assm: h4/numRing/num__ring__thms_c29: !t l0 l c0 c. h4/numRing/num__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__monom c l t) = h4/numRing/num__monom__insert (h4/arithmetic/_2A c0 c) (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/numRing/num__canonical__sum__scalar3 c0 l0 t)
% Assm: h4/numRing/num__ring__thms_c30: !t l0 l c0. h4/numRing/num__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__varlist l t) = h4/numRing/num__monom__insert c0 (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/numRing/num__canonical__sum__scalar3 c0 l0 t)
% Assm: h4/integerRing/int__ring__thms_c31: !t l0 l c0. h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__monom__insert c0 (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar3 c0 l0 t)
% Assm: h4/integerRing/int__ring__thms_c28: !t l0 l. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar2 l0 t)
% Assm: h4/numRing/num__ring__thms_c27: !t l0 l. h4/numRing/num__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/numRing/num__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/numRing/num__canonical__sum__scalar2 l0 t)
% Assm: h4/canonical/varlist__merge__ok: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm x y. h4/canonical/interp__vl sr vm (h4/prelim/list__merge h4/quote/index__lt x y) = h4/semi__ring/semi__ring__SRM sr (h4/canonical/interp__vl sr vm x) (h4/canonical/interp__vl sr vm y))
% Assm: h4/ratRing/rat__ring__thms_c20: !l1 c1. h4/ratRing/rat__r__monom__insert c1 l1 h4/canonical/Nil__monom = h4/canonical/Cons__monom c1 l1 h4/canonical/Nil__monom
% Assm: h4/ratRing/rat__r__canonical__sum__scalar__def: h4/ratRing/rat__r__canonical__sum__scalar = h4/ringNorm/r__canonical__sum__scalar (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__scalar__def_c2: !r c0. h4/ringNorm/r__canonical__sum__scalar r c0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/ringNorm/canonical__sum__scalar__def_c0: !t r l c0 c. h4/ringNorm/r__canonical__sum__scalar r c0 (h4/canonical/Cons__monom c l t) = h4/canonical/Cons__monom (h4/ring/ring__RM r c0 c) l (h4/ringNorm/r__canonical__sum__scalar r c0 t)
% Assm: h4/ringNorm/canonical__sum__scalar__def_c1: !t r l c0. h4/ringNorm/r__canonical__sum__scalar r c0 (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__monom c0 l (h4/ringNorm/r__canonical__sum__scalar r c0 t)
% Assm: h4/ringNorm/r__varlist__insert__def: !r. h4/ringNorm/r__varlist__insert r = h4/canonical/varlist__insert (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__canonical__sum__scalar2__def: !r. h4/ringNorm/r__canonical__sum__scalar2 r = h4/canonical/canonical__sum__scalar2 (h4/ring/semi__ring__of r)
% Assm: h4/numRing/num__varlist__insert__def: h4/numRing/num__varlist__insert = h4/canonical/varlist__insert (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__canonical__sum__scalar2__def: h4/numRing/num__canonical__sum__scalar2 = h4/canonical/canonical__sum__scalar2 (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/integerRing/int__r__canonical__sum__scalar2__def: h4/integerRing/int__r__canonical__sum__scalar2 = h4/ringNorm/r__canonical__sum__scalar2 (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/integerRing/int__r__varlist__insert__def: h4/integerRing/int__r__varlist__insert = h4/ringNorm/r__varlist__insert (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/integerRing/int__r__canonical__sum__scalar3__def: h4/integerRing/int__r__canonical__sum__scalar3 = h4/ringNorm/r__canonical__sum__scalar3 (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/ringNorm/canonical__sum__scalar3__def_c2: !r l0 c0. h4/ringNorm/r__canonical__sum__scalar3 r c0 l0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/numRing/num__canonical__sum__scalar3__def: h4/numRing/num__canonical__sum__scalar3 = h4/canonical/canonical__sum__scalar3 (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/ring/semi__ring__of__def: !r. h4/ring/semi__ring__of r = h4/semi__ring/semi__ring0 (h4/ring/ring__R0 r) (h4/ring/ring__R1 r) (h4/ring/ring__RP r) (h4/ring/ring__RM r)
% Assm: h4/ringNorm/r__canonical__sum__scalar3__def: !r. h4/ringNorm/r__canonical__sum__scalar3 r = h4/canonical/canonical__sum__scalar3 (h4/ring/semi__ring__of r)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% 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/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/canonical/hidden____20__40ind____typecanonical0____def: h4/canonical/_20_40ind__typecanonical0 = h4/canonical/_20_40ind__typecanonical3 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (\n. h4/ind__type/BOTTOM))
% Assm: h4/canonical/hidden____20__40ind____typecanonical1____def: h4/canonical/_20_40ind__typecanonical1 = (\a0 a1 a2. h4/canonical/_20_40ind__typecanonical3 ((\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 a10) (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM))) a0 a1 (h4/canonical/_20_40ind__typecanonical4 a2)))
% Assm: h4/canonical/Cons__varlist0: h4/canonical/Cons__varlist = h4/canonical/_20_40ind__typecanonical2
% Assm: h4/canonical/Cons__monom0: h4/canonical/Cons__monom = h4/canonical/_20_40ind__typecanonical1
% Assm: h4/canonical/Nil__monom0: h4/canonical/Nil__monom = h4/canonical/_20_40ind__typecanonical0
% Assm: h4/canonical/canonical__sum__repfns_c1: !r. (\a0_27. !_27canonical__sum_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (\n. h4/ind__type/BOTTOM) \/ (?a0 a1 a2. a0_270 = (\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 a10) (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM))) a0 a1 a2 /\ _27canonical__sum_27 a2) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB a00) (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27canonical__sum_27 a1) ==> _27canonical__sum_27 a0_270) ==> _27canonical__sum_27 a0_27) r <=> h4/canonical/_20_40ind__typecanonical4 (h4/canonical/_20_40ind__typecanonical3 r) = r
% Assm: h4/canonical/hidden____20__40ind____typecanonical2____def: h4/canonical/_20_40ind__typecanonical2 = (\a0 a1. h4/canonical/_20_40ind__typecanonical3 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB a00) (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/canonical/_20_40ind__typecanonical4 a1)))
% Assm: h4/canonical/canonical__sum__repfns_c0: !a. h4/canonical/_20_40ind__typecanonical3 (h4/canonical/_20_40ind__typecanonical4 a) = a
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/ratRing/rat__ring__thms_c25: !t l c0. h4/ratRing/rat__r__canonical__sum__scalar c0 (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__monom c0 l (h4/ratRing/rat__r__canonical__sum__scalar c0 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_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_c17: !v8 v7. h4/ratRing/rat__r__canonical__sum__merge h4/canonical/Nil__monom (h4/canonical/Cons__varlist v7 v8) = h4/canonical/Cons__varlist v7 v8
% Assm: h4/ratRing/rat__ring__thms_c15: !s1. h4/ratRing/rat__r__canonical__sum__merge s1 h4/canonical/Nil__monom = s1
% Assm: h4/rat/RAT__SUB__RDISTRIB: !c b a. h4/rat/rat__mul (h4/rat/rat__sub a b) c = h4/rat/rat__sub (h4/rat/rat__mul a c) (h4/rat/rat__mul b c)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/canonical/interp__vl__def_c0: !vm sr. h4/canonical/interp__vl sr vm h4/list/NIL = h4/semi__ring/semi__ring__SR1 sr
% Assm: h4/canonical/interp__vl__def_c1: !x vm t sr. h4/canonical/interp__vl sr vm (h4/list/CONS x t) = h4/canonical/ivl__aux sr vm x t
% Assm: h4/canonical/ivl__aux__def_c1: !x_27 x vm t_27 sr. h4/canonical/ivl__aux sr vm x (h4/list/CONS x_27 t_27) = h4/semi__ring/semi__ring__SRM sr (h4/quote/varmap__find x vm) (h4/canonical/ivl__aux sr vm x_27 t_27)
% Assm: h4/canonical/ivl__aux__def_c0: !x vm sr. h4/canonical/ivl__aux sr vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/quote/index__lt__def: !i2 i1. h4/quote/index__lt i1 i2 <=> h4/quote/index__compare i1 i2 = h4/prelim/LESS
% Assm: h4/semi__ring/mult__one__right: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRM r n (h4/semi__ring/semi__ring__SR1 r) = n)
% Assm: h4/semi__ring/mult__rotate: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r m n) p = h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r n p) m)
% Assm: h4/semi__ring/mult__one__left: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SR1 r) n = n)
% Assm: h4/prelim/list__merge__def_c0: !l1 a__lt. h4/prelim/list__merge a__lt l1 h4/list/NIL = l1
% Assm: h4/semi__ring/mult__assoc: !r. h4/semi__ring/is__semi__ring r ==> (!n m p. h4/semi__ring/semi__ring__SRM r n (h4/semi__ring/semi__ring__SRM r m p) = h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r n m) p)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/prelim/list__merge__def_c1: !v5 v4 a__lt. h4/prelim/list__merge a__lt h4/list/NIL (h4/list/CONS v4 v5) = h4/list/CONS v4 v5
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/prelim/list__merge__def_c2: !y x l2 l1 a__lt. h4/prelim/list__merge a__lt (h4/list/CONS x l1) (h4/list/CONS y l2) = h4/bool/COND (a__lt x y) (h4/list/CONS x (h4/prelim/list__merge a__lt l1 (h4/list/CONS y l2))) (h4/list/CONS y (h4/prelim/list__merge a__lt (h4/list/CONS x l1) l2))
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/rat/RAT__AINV__LMUL: !r2 r1. h4/rat/rat__ainv (h4/rat/rat__mul r1 r2) = h4/rat/rat__mul (h4/rat/rat__ainv r1) r2
% Assm: h4/rat/RAT__SUB__ADDAINV: !r2 r1. h4/rat/rat__sub r1 r2 = h4/rat/rat__add r1 (h4/rat/rat__ainv r2)
% Assm: h4/rat/RAT__RDISTRIB: !c b a. h4/rat/rat__mul (h4/rat/rat__add a b) c = h4/rat/rat__add (h4/rat/rat__mul a c) (h4/rat/rat__mul b c)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Goal: !t l0 l c0 c. h4/ratRing/rat__r__canonical__sum__scalar3 c0 l0 (h4/canonical/Cons__monom c l t) = h4/ratRing/rat__r__monom__insert (h4/rat/rat__mul c0 c) (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/ratRing/rat__r__canonical__sum__scalar3 c0 l0 t)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_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_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_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_ringNorms_canonicalu_u_sumu_u_scalar2u_u_defu_c0]: !t r l0 l c. happ (happ (h4/ringNorm/r__canonical__sum__scalar2 r) l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ (h4/ringNorm/r__monom__insert r) c) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (h4/ringNorm/r__canonical__sum__scalar2 r) l0) t)
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_scalar2u_u_defu_c1]: !t r l0 l. happ (happ (h4/ringNorm/r__canonical__sum__scalar2 r) l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ (h4/ringNorm/r__varlist__insert r) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (h4/ringNorm/r__canonical__sum__scalar2 r) l0) t)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c27]: !t l0 l c. happ (happ h4/ratRing/rat__r__canonical__sum__scalar2 l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ h4/ratRing/rat__r__monom__insert c) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ h4/ratRing/rat__r__canonical__sum__scalar2 l0) t)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalar2u_u_defu_c0]: !t sr l0 l c. happ (happ (h4/canonical/canonical__sum__scalar2 sr) l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ (h4/canonical/monom__insert sr) c) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (h4/canonical/canonical__sum__scalar2 sr) l0) t)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalar2u_u_defu_c1]: !t sr l0 l. happ (happ (h4/canonical/canonical__sum__scalar2 sr) l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ (h4/canonical/varlist__insert sr) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (h4/canonical/canonical__sum__scalar2 sr) l0) t)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalar3u_u_defu_c0]: !t sr l0 l c0 c. happ (happ (happ (h4/canonical/canonical__sum__scalar3 sr) c0) l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ (h4/canonical/monom__insert sr) (happ (happ (h4/semi__ring/semi__ring__SRM sr) c0) c)) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ (h4/canonical/canonical__sum__scalar3 sr) c0) l0) t)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalar3u_u_defu_c1]: !t sr l0 l c0. happ (happ (happ (h4/canonical/canonical__sum__scalar3 sr) c0) l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ (happ (h4/canonical/monom__insert sr) c0) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ (h4/canonical/canonical__sum__scalar3 sr) c0) l0) t)
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_scalar2u_u_defu_c2]: !r l0. happ (happ (h4/ringNorm/r__canonical__sum__scalar2 r) l0) h4/canonical/Nil__monom = h4/canonical/Nil__monom
% 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_ringu_u_thmsu_c18]: !t2 l2 l1 c2 c1. happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ (happ h4/canonical/Cons__monom c1) l1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2)) (happ (happ (happ h4/canonical/Cons__monom (happ (happ h4/rat/rat__add c1) c2)) l1) t2) (happ (happ (happ h4/canonical/Cons__monom c2) l2) (happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) t2))
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_scalar3u_u_defu_c0]: !t r l0 l c0 c. happ (happ (happ (h4/ringNorm/r__canonical__sum__scalar3 r) c0) l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ (h4/ringNorm/r__monom__insert r) (happ (happ (h4/ring/ring__RM r) c0) c)) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ (h4/ringNorm/r__canonical__sum__scalar3 r) c0) l0) t)
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_scalar3u_u_defu_c1]: !t r l0 l c0. happ (happ (happ (h4/ringNorm/r__canonical__sum__scalar3 r) c0) l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ (happ (h4/ringNorm/r__monom__insert r) c0) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ (h4/ringNorm/r__canonical__sum__scalar3 r) c0) l0) t)
% 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_ringNorms_monomu_u_insertu_u_defu_c0]: !t2 r l2 l1 c2 c1. happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ (happ h4/canonical/Cons__monom c1) l1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2)) (happ (happ (happ h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) c2)) l1) t2) (happ (happ (happ 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_c2]: !r l1 c1. happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) h4/canonical/Nil__monom = happ (happ (happ h4/canonical/Cons__monom c1) l1) h4/canonical/Nil__monom
% 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) (happ (happ h4/canonical/Cons__varlist l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ (happ h4/canonical/Cons__monom c1) l1) (happ (happ h4/canonical/Cons__varlist l2) t2)) (happ (happ (happ h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) (h4/ring/ring__R1 r))) l1) t2) (happ (happ h4/canonical/Cons__varlist l2) (happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) t2))
% 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_canonicals_canonicalu_u_sumu_u_scalar2u_u_defu_c2]: !sr l0. happ (happ (h4/canonical/canonical__sum__scalar2 sr) l0) h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_numRings_numu_u_monomu_u_insertu_u_def]: h4/numRing/num__monom__insert = h4/canonical/monom__insert (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_integerRings_intu_u_ru_u_monomu_u_insertu_u_def]: h4/integerRing/int__r__monom__insert = h4/ringNorm/r__monom__insert (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_ringNorms_ru_u_monomu_u_insertu_u_def]: !r. h4/ringNorm/r__monom__insert r = h4/canonical/monom__insert (h4/ring/semi__ring__of r)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalar3u_u_defu_c2]: !sr l0 c0. happ (happ (happ (h4/canonical/canonical__sum__scalar3 sr) c0) l0) h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c28]: !t l0 l. happ (happ h4/ratRing/rat__r__canonical__sum__scalar2 l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ h4/ratRing/rat__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ h4/ratRing/rat__r__canonical__sum__scalar2 l0) t)
% 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) (happ (happ (happ h4/canonical/Cons__monom c1) l1) t1)) (happ (happ h4/canonical/Cons__varlist l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ (happ h4/canonical/Cons__monom c1) l1) (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (happ (happ h4/canonical/Cons__varlist l2) t2))) (happ (happ (happ 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)) (happ (happ h4/canonical/Cons__varlist l2) (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (happ (happ (happ h4/canonical/Cons__monom c1) l1) t1)) t2))
% 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) (happ (happ h4/canonical/Cons__varlist l1) t1)) (happ (happ h4/canonical/Cons__varlist l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ h4/canonical/Cons__varlist l1) (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (happ (happ h4/canonical/Cons__varlist l2) t2))) (happ (happ (happ 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)) (happ (happ h4/canonical/Cons__varlist l2) (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (happ (happ h4/canonical/Cons__varlist 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_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) (happ (happ h4/canonical/Cons__varlist l1) t1)) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ h4/canonical/Cons__varlist l1) (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2))) (happ (happ (happ 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)) (happ (happ (happ h4/canonical/Cons__monom c2) l2) (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (happ (happ 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) (happ (happ (happ h4/canonical/Cons__monom c1) l1) t1)) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ (happ h4/canonical/Cons__monom c1) l1) (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2))) (happ (happ (happ 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)) (happ (happ (happ h4/canonical/Cons__monom c2) l2) (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (happ (happ (happ h4/canonical/Cons__monom c1) l1) t1)) t2))
% 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) (happ (happ (happ h4/canonical/Cons__monom v4) v5) v6) = happ (happ (happ h4/canonical/Cons__monom v4) v5) v6
% 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) (happ (happ h4/canonical/Cons__varlist v7) v8) = happ (happ h4/canonical/Cons__varlist v7) v8
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c24]: !t l c0 c. happ (happ h4/ratRing/rat__r__canonical__sum__scalar c0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ h4/canonical/Cons__monom (happ (happ h4/rat/rat__mul c0) c)) l) (happ (happ h4/ratRing/rat__r__canonical__sum__scalar c0) t)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c19]: !t2 l2 l1 c1. happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) (happ (happ h4/canonical/Cons__varlist l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ (happ h4/canonical/Cons__monom c1) l1) (happ (happ h4/canonical/Cons__varlist l2) t2)) (happ (happ (happ 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) (happ (happ h4/canonical/Cons__varlist l2) (happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c21]: !t2 l2 l1 c2. happ (happ h4/ratRing/rat__r__varlist__insert l1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ h4/canonical/Cons__varlist l1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2)) (happ (happ (happ 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) (happ (happ (happ h4/canonical/Cons__monom c2) l2) (happ (happ h4/ratRing/rat__r__varlist__insert l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c11]: !t2 t1 l2 l1 c2 c1. happ (happ h4/ratRing/rat__r__canonical__sum__merge (happ (happ (happ h4/canonical/Cons__monom c1) l1) t1)) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ (happ h4/canonical/Cons__monom c1) l1) (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2))) (happ (happ (happ h4/canonical/Cons__monom (happ (happ h4/rat/rat__add c1) c2)) l1) (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) t2)) (happ (happ (happ h4/canonical/Cons__monom c2) l2) (happ (happ h4/ratRing/rat__r__canonical__sum__merge (happ (happ (happ h4/canonical/Cons__monom c1) l1) t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c16]: !v6 v5 v4. happ (happ h4/ratRing/rat__r__canonical__sum__merge h4/canonical/Nil__monom) (happ (happ (happ h4/canonical/Cons__monom v4) v5) v6) = happ (happ (happ h4/canonical/Cons__monom v4) v5) v6
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c12]: !t2 t1 l2 l1 c1. happ (happ h4/ratRing/rat__r__canonical__sum__merge (happ (happ (happ h4/canonical/Cons__monom c1) l1) t1)) (happ (happ h4/canonical/Cons__varlist l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ (happ h4/canonical/Cons__monom c1) l1) (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (happ (happ h4/canonical/Cons__varlist l2) t2))) (happ (happ (happ 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)) (happ (happ h4/canonical/Cons__varlist l2) (happ (happ h4/ratRing/rat__r__canonical__sum__merge (happ (happ (happ 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 (happ (happ h4/canonical/Cons__varlist l1) t1)) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ h4/canonical/Cons__varlist l1) (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2))) (happ (happ (happ 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)) (happ (happ (happ h4/canonical/Cons__monom c2) l2) (happ (happ h4/ratRing/rat__r__canonical__sum__merge (happ (happ h4/canonical/Cons__varlist l1) t1)) t2))
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c26]: !t l0 l c. happ (happ h4/numRing/num__canonical__sum__scalar2 l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ h4/numRing/num__monom__insert c) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ h4/numRing/num__canonical__sum__scalar2 l0) t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c27]: !t l0 l c. happ (happ h4/integerRing/int__r__canonical__sum__scalar2 l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ h4/integerRing/int__r__monom__insert c) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ h4/integerRing/int__r__canonical__sum__scalar2 l0) t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c30]: !t l0 l c0 c. happ (happ (happ h4/integerRing/int__r__canonical__sum__scalar3 c0) l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ h4/integerRing/int__r__monom__insert (happ (happ h4/integer/int__mul c0) c)) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ h4/integerRing/int__r__canonical__sum__scalar3 c0) l0) t)
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c29]: !t l0 l c0 c. happ (happ (happ h4/numRing/num__canonical__sum__scalar3 c0) l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ h4/numRing/num__monom__insert (happ (happ h4/arithmetic/_2A c0) c)) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ h4/numRing/num__canonical__sum__scalar3 c0) l0) t)
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c30]: !t l0 l c0. happ (happ (happ h4/numRing/num__canonical__sum__scalar3 c0) l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ (happ h4/numRing/num__monom__insert c0) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ h4/numRing/num__canonical__sum__scalar3 c0) l0) t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c31]: !t l0 l c0. happ (happ (happ h4/integerRing/int__r__canonical__sum__scalar3 c0) l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ (happ h4/integerRing/int__r__monom__insert c0) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ h4/integerRing/int__r__canonical__sum__scalar3 c0) l0) t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c28]: !t l0 l. happ (happ h4/integerRing/int__r__canonical__sum__scalar2 l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ h4/integerRing/int__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ h4/integerRing/int__r__canonical__sum__scalar2 l0) t)
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c27]: !t l0 l. happ (happ h4/numRing/num__canonical__sum__scalar2 l0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ h4/numRing/num__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ h4/numRing/num__canonical__sum__scalar2 l0) t)
% Assm [h4s_canonicals_varlistu_u_mergeu_u_ok]: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm x y. h4/canonical/interp__vl sr vm (h4/prelim/list__merge h4/quote/index__lt x y) = happ (happ (h4/semi__ring/semi__ring__SRM sr) (h4/canonical/interp__vl sr vm x)) (h4/canonical/interp__vl sr vm y))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c20]: !l1 c1. happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) h4/canonical/Nil__monom = happ (happ (happ h4/canonical/Cons__monom c1) l1) h4/canonical/Nil__monom
% Assm [h4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_scalaru_u_def]: h4/ratRing/rat__r__canonical__sum__scalar = h4/ringNorm/r__canonical__sum__scalar (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_scalaru_u_defu_c2]: !r c0. happ (happ (h4/ringNorm/r__canonical__sum__scalar r) c0) h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_scalaru_u_defu_c0]: !t r l c0 c. happ (happ (h4/ringNorm/r__canonical__sum__scalar r) c0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RM r) c0) c)) l) (happ (happ (h4/ringNorm/r__canonical__sum__scalar r) c0) t)
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_scalaru_u_defu_c1]: !t r l c0. happ (happ (h4/ringNorm/r__canonical__sum__scalar r) c0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ (happ h4/canonical/Cons__monom c0) l) (happ (happ (h4/ringNorm/r__canonical__sum__scalar r) c0) t)
% Assm [h4s_ringNorms_ru_u_varlistu_u_insertu_u_def]: !r. h4/ringNorm/r__varlist__insert r = h4/canonical/varlist__insert (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_canonicalu_u_sumu_u_scalar2u_u_def]: !r. h4/ringNorm/r__canonical__sum__scalar2 r = h4/canonical/canonical__sum__scalar2 (h4/ring/semi__ring__of r)
% Assm [h4s_numRings_numu_u_varlistu_u_insertu_u_def]: h4/numRing/num__varlist__insert = h4/canonical/varlist__insert (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_scalar2u_u_def]: h4/numRing/num__canonical__sum__scalar2 = h4/canonical/canonical__sum__scalar2 (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2u_u_def]: h4/integerRing/int__r__canonical__sum__scalar2 = h4/ringNorm/r__canonical__sum__scalar2 (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_integerRings_intu_u_ru_u_varlistu_u_insertu_u_def]: h4/integerRing/int__r__varlist__insert = h4/ringNorm/r__varlist__insert (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3u_u_def]: h4/integerRing/int__r__canonical__sum__scalar3 = h4/ringNorm/r__canonical__sum__scalar3 (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_scalar3u_u_defu_c2]: !r l0 c0. happ (happ (happ (h4/ringNorm/r__canonical__sum__scalar3 r) c0) l0) h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_scalar3u_u_def]: h4/numRing/num__canonical__sum__scalar3 = h4/canonical/canonical__sum__scalar3 (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_rings_semiu_u_ringu_u_ofu_u_def]: !r. h4/ring/semi__ring__of r = h4/semi__ring/semi__ring0 (h4/ring/ring__R0 r) (h4/ring/ring__R1 r) (h4/ring/ring__RP r) (h4/ring/ring__RM r)
% Assm [h4s_ringNorms_ru_u_canonicalu_u_sumu_u_scalar3u_u_def]: !r. h4/ringNorm/r__canonical__sum__scalar3 r = h4/canonical/canonical__sum__scalar3 (h4/ring/semi__ring__of r)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_ringNorms_varlistu_u_insertu_u_defu_c1]: !t2 r l2 l1. happ (happ (h4/ringNorm/r__varlist__insert r) l1) (happ (happ h4/canonical/Cons__varlist l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ h4/canonical/Cons__varlist l1) (happ (happ h4/canonical/Cons__varlist l2) t2)) (happ (happ (happ h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) (h4/ring/ring__R1 r))) l1) t2) (happ (happ 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 = happ (happ 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) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ h4/canonical/Cons__varlist l1) (happ (happ (happ h4/canonical/Cons__monom c2) l2) t2)) (happ (happ (happ h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) c2)) l1) t2) (happ (happ (happ h4/canonical/Cons__monom c2) l2) (happ (happ (h4/ringNorm/r__varlist__insert r) l1) t2))
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/canonical/_20_40ind__typecanonical0 = h4/canonical/_20_40ind__typecanonical3 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) _0)
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x x. happ (happ (happ h4/canonical/_20_40ind__typecanonical1 x) x) x = h4/canonical/_20_40ind__typecanonical3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C x x) (h4/ind__type/FCONS (h4/canonical/_20_40ind__typecanonical4 x) _0)))
% Assm [h4s_canonicals_Consu_u_varlist0]: h4/canonical/Cons__varlist = h4/canonical/_20_40ind__typecanonical2
% Assm [h4s_canonicals_Consu_u_monom0]: h4/canonical/Cons__monom = h4/canonical/_20_40ind__typecanonical1
% Assm [h4s_canonicals_Nilu_u_monom0]: h4/canonical/Nil__monom = h4/canonical/_20_40ind__typecanonical0
% Assm [h4s_canonicals_canonicalu_u_sumu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27canonical__sum_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) _0 \/ (?a0 a1 a2. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a0 a1) (h4/ind__type/FCONS a2 _0) /\ happ _27canonical__sum_27 a2) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB a0) (h4/ind__type/FCONS a1 _0) /\ happ _27canonical__sum_27 a1) ==> happ _27canonical__sum_27 a0_270) ==> happ _27canonical__sum_27 r) <=> h4/canonical/_20_40ind__typecanonical4 (h4/canonical/_20_40ind__typecanonical3 r) = r)
% Assm [h4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/canonical/_20_40ind__typecanonical2 x) x = h4/canonical/_20_40ind__typecanonical3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB x) (h4/ind__type/FCONS (h4/canonical/_20_40ind__typecanonical4 x) _0)))
% Assm [h4s_canonicals_canonicalu_u_sumu_u_repfnsu_c0]: !a. h4/canonical/_20_40ind__typecanonical3 (h4/canonical/_20_40ind__typecanonical4 a) = a
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c25]: !t l c0. happ (happ h4/ratRing/rat__r__canonical__sum__scalar c0) (happ (happ h4/canonical/Cons__varlist l) t) = happ (happ (happ h4/canonical/Cons__monom c0) l) (happ (happ h4/ratRing/rat__r__canonical__sum__scalar c0) t)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c22]: !t2 l2 l1. happ (happ h4/ratRing/rat__r__varlist__insert l1) (happ (happ h4/canonical/Cons__varlist l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ h4/canonical/Cons__varlist l1) (happ (happ h4/canonical/Cons__varlist l2) t2)) (happ (happ (happ 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) (happ (happ h4/canonical/Cons__varlist l2) (happ (happ h4/ratRing/rat__r__varlist__insert l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c14]: !t2 t1 l2 l1. happ (happ h4/ratRing/rat__r__canonical__sum__merge (happ (happ h4/canonical/Cons__varlist l1) t1)) (happ (happ h4/canonical/Cons__varlist l2) t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (happ (happ h4/canonical/Cons__varlist l1) (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (happ (happ h4/canonical/Cons__varlist l2) t2))) (happ (happ (happ 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)) (happ (happ h4/canonical/Cons__varlist l2) (happ (happ h4/ratRing/rat__r__canonical__sum__merge (happ (happ h4/canonical/Cons__varlist l1) t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c17]: !v8 v7. happ (happ h4/ratRing/rat__r__canonical__sum__merge h4/canonical/Nil__monom) (happ (happ h4/canonical/Cons__varlist v7) v8) = happ (happ h4/canonical/Cons__varlist v7) v8
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c15]: !s1. happ (happ h4/ratRing/rat__r__canonical__sum__merge s1) h4/canonical/Nil__monom = s1
% Assm [h4s_rats_RATu_u_SUBu_u_RDISTRIB]: !c b a. happ (happ h4/rat/rat__mul (h4/rat/rat__sub a b)) c = h4/rat/rat__sub (happ (happ h4/rat/rat__mul a) c) (happ (happ h4/rat/rat__mul b) c)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_canonicals_interpu_u_vlu_u_defu_c0]: !vm sr. h4/canonical/interp__vl sr vm h4/list/NIL = h4/semi__ring/semi__ring__SR1 sr
% Assm [h4s_canonicals_interpu_u_vlu_u_defu_c1]: !x vm t sr. h4/canonical/interp__vl sr vm (h4/list/CONS x t) = h4/canonical/ivl__aux sr vm x t
% Assm [h4s_canonicals_ivlu_u_auxu_u_defu_c1]: !x_27 x vm t_27 sr. h4/canonical/ivl__aux sr vm x (h4/list/CONS x_27 t_27) = happ (happ (h4/semi__ring/semi__ring__SRM sr) (h4/quote/varmap__find x vm)) (h4/canonical/ivl__aux sr vm x_27 t_27)
% Assm [h4s_canonicals_ivlu_u_auxu_u_defu_c0]: !x vm sr. h4/canonical/ivl__aux sr vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_quotes_indexu_u_ltu_u_def]: !i2 i1. happ (happ h4/quote/index__lt i1) i2 <=> happ (happ h4/quote/index__compare i1) i2 = h4/prelim/LESS
% Assm [h4s_semiu_u_rings_multu_u_oneu_u_right]: !r. h4/semi__ring/is__semi__ring r ==> (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) n) (h4/semi__ring/semi__ring__SR1 r) = n)
% Assm [h4s_semiu_u_rings_multu_u_rotate]: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. happ (happ (h4/semi__ring/semi__ring__SRM r) (happ (happ (h4/semi__ring/semi__ring__SRM r) m) n)) p = happ (happ (h4/semi__ring/semi__ring__SRM r) (happ (happ (h4/semi__ring/semi__ring__SRM r) n) p)) m)
% Assm [h4s_semiu_u_rings_multu_u_oneu_u_left]: !r. h4/semi__ring/is__semi__ring r ==> (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) (h4/semi__ring/semi__ring__SR1 r)) n = n)
% Assm [h4s_prelims_listu_u_mergeu_u_defu_c0]: !l1 a__lt. h4/prelim/list__merge a__lt l1 h4/list/NIL = l1
% Assm [h4s_semiu_u_rings_multu_u_assoc]: !r. h4/semi__ring/is__semi__ring r ==> (!n m p. happ (happ (h4/semi__ring/semi__ring__SRM r) n) (happ (happ (h4/semi__ring/semi__ring__SRM r) m) p) = happ (happ (h4/semi__ring/semi__ring__SRM r) (happ (happ (h4/semi__ring/semi__ring__SRM r) n) m)) p)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_prelims_listu_u_mergeu_u_defu_c1]: !v5 v4 a__lt. h4/prelim/list__merge a__lt h4/list/NIL (h4/list/CONS v4 v5) = h4/list/CONS v4 v5
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_prelims_listu_u_mergeu_u_defu_c2]: !y x l2 l1 a__lt. h4/prelim/list__merge a__lt (h4/list/CONS x l1) (h4/list/CONS y l2) = h4/bool/COND (happ (happ a__lt x) y) (h4/list/CONS x (h4/prelim/list__merge a__lt l1 (h4/list/CONS y l2))) (h4/list/CONS y (h4/prelim/list__merge a__lt (h4/list/CONS x l1) l2))
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_rats_RATu_u_AINVu_u_LMUL]: !r2 r1. happ h4/rat/rat__ainv (happ (happ h4/rat/rat__mul r1) r2) = happ (happ h4/rat/rat__mul (happ h4/rat/rat__ainv r1)) r2
% Assm [h4s_rats_RATu_u_SUBu_u_ADDAINV]: !r2 r1. h4/rat/rat__sub r1 r2 = happ (happ h4/rat/rat__add r1) (happ h4/rat/rat__ainv r2)
% Assm [h4s_rats_RATu_u_RDISTRIB]: !c b a. happ (happ h4/rat/rat__mul (happ (happ h4/rat/rat__add a) b)) c = happ (happ h4/rat/rat__add (happ (happ h4/rat/rat__mul a) c)) (happ (happ h4/rat/rat__mul b) c)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Goal: !t l0 l c0 c. happ (happ (happ h4/ratRing/rat__r__canonical__sum__scalar3 c0) l0) (happ (happ (happ h4/canonical/Cons__monom c) l) t) = happ (happ (happ h4/ratRing/rat__r__monom__insert (happ (happ h4/rat/rat__mul c0) c)) (h4/prelim/list__merge h4/quote/index__lt l0 l)) (happ (happ (happ h4/ratRing/rat__r__canonical__sum__scalar3 c0) l0) t)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1330762,TV_Q1330758]: ![V_f, V_g]: (![V_x]: s(TV_Q1330758,happ(s(t_fun(TV_Q1330762,TV_Q1330758),V_f),s(TV_Q1330762,V_x))) = s(TV_Q1330758,happ(s(t_fun(TV_Q1330762,TV_Q1330758),V_g),s(TV_Q1330762,V_x))) => s(t_fun(TV_Q1330762,TV_Q1330758),V_f) = s(t_fun(TV_Q1330762,TV_Q1330758),V_g))).
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_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_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_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_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_ringNorms_canonicalu_u_sumu_u_scalar2u_u_defu_c0, axiom, ![TV_u_27a]: ![V_t, V_r, V_l0, V_l, V_c]: 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_canonicalu_u_sumu_u_scalar2(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(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_canonicalu_u_sumu_u_scalar2(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_scalar2u_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_r, V_l0, 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)),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_canonicalu_u_sumu_u_scalar2(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(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_canonicalu_u_sumu_u_scalar2(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c27, axiom, ![V_t, V_l0, V_l, V_c]: 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_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_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_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),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_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_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_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t)))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalar2u_u_defu_c0, axiom, ![TV_u_27a]: ![V_t, V_sr, V_l0, V_l, V_c]: 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_canonicals_canonicalu_u_sumu_u_scalar2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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_canonicals_monomu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(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_canonicals_canonicalu_u_sumu_u_scalar2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalar2u_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_sr, V_l0, 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)),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_canonicals_canonicalu_u_sumu_u_scalar2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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_canonicals_varlistu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(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_canonicals_canonicalu_u_sumu_u_scalar2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalar3u_u_defu_c0, axiom, ![TV_u_27a]: ![V_t, V_sr, V_l0, V_l, V_c0, V_c]: 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_canonicals_canonicalu_u_sumu_u_scalar3(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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_canonicals_monomu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(TV_u_27a,V_c))))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(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_canonicals_canonicalu_u_sumu_u_scalar3(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalar3u_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_sr, V_l0, V_l, V_c0]: 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_canonicals_canonicalu_u_sumu_u_scalar3(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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_canonicals_monomu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(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_canonicals_canonicalu_u_sumu_u_scalar3(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_scalar2u_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_l0]: 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_canonicalu_u_sumu_u_scalar2(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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_ringu_u_thmsu_c18, axiom, ![V_t2, V_l2, V_l1, V_c2, 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),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_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),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_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_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_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),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_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,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),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_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))),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_ringNorms_canonicalu_u_sumu_u_scalar3u_u_defu_c0, axiom, ![TV_u_27a]: ![V_t, V_r, V_l0, V_l, V_c0, V_c]: 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_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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,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,V_c0))),s(TV_u_27a,V_c))))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(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_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_scalar3u_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_r, V_l0, V_l, V_c0]: 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_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(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_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
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_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),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_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),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_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_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_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),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_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),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_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_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),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_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_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),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_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),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_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_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_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),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_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),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_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_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_canonicals_canonicalu_u_sumu_u_scalar2u_u_defu_c2, axiom, ![TV_u_27a]: ![V_sr, V_l0]: 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_canonicals_canonicalu_u_sumu_u_scalar2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_numRings_numu_u_monomu_u_insertu_u_def, axiom, s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_monomu_u_insert) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_canonicals_monomu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_integerRings_intu_u_ru_u_monomu_u_insertu_u_def, axiom, s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_monomu_u_insert) = s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_ringNorms_ru_u_monomu_u_insertu_u_def, axiom, ![TV_u_27a]: ![V_r]: 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(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_canonicals_monomu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalar3u_u_defu_c2, axiom, ![TV_u_27a]: ![V_sr, V_l0, V_c0]: 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_canonicals_canonicalu_u_sumu_u_scalar3(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_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,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c28, axiom, ![V_t, V_l0, 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)),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_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_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_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),V_t))))) = 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),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_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_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t)))))).
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),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_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),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_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),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_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),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_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),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_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),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_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),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_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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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_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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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_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),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_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),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_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_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),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_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),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_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_ratRings_ratu_u_ringu_u_thmsu_c24, axiom, ![V_t, V_l, V_c0, V_c]: 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_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_scalar),s(t_h4s_rats_rat,V_c0))),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_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),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_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_mul),s(t_h4s_rats_rat,V_c0))),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)),happ(s(t_fun(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_scalar),s(t_h4s_rats_rat,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t)))))).
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),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_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),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_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_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_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),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_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),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_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_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),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_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),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_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_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_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),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_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),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_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_c11, axiom, ![V_t2, V_t1, V_l2, V_l1, V_c2, 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),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_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),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_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),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_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),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_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),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_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,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),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_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),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_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_c16, axiom, ![V_v6, V_v5, V_v4]: 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_nilu_u_monom))),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_canonicals_consu_u_monom),s(t_h4s_rats_rat,V_v4))),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_v6))))) = 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_canonicals_consu_u_monom),s(t_h4s_rats_rat,V_v4))),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_v6)))).
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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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_numRings_numu_u_ringu_u_thmsu_c26, axiom, ![V_t, V_l0, V_l, V_c]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_canonicals_consu_u_monom),s(t_h4s_nums_num,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_monomu_u_insert),s(t_h4s_nums_num,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c27, axiom, ![V_t, V_l0, V_l, V_c]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_canonicals_consu_u_monom),s(t_h4s_integers_int,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_monomu_u_insert),s(t_h4s_integers_int,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c30, axiom, ![V_t, V_l0, V_l, V_c0, V_c]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3),s(t_h4s_integers_int,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_canonicals_consu_u_monom),s(t_h4s_integers_int,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_monomu_u_insert),s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_h4s_integers_int,V_c0))),s(t_h4s_integers_int,V_c))))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3),s(t_h4s_integers_int,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c29, axiom, ![V_t, V_l0, V_l, V_c0, V_c]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar3),s(t_h4s_nums_num,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_canonicals_consu_u_monom),s(t_h4s_nums_num,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_monomu_u_insert),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,V_c0))),s(t_h4s_nums_num,V_c))))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar3),s(t_h4s_nums_num,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t)))))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c30, axiom, ![V_t, V_l0, V_l, V_c0]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar3),s(t_h4s_nums_num,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),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_nums_num),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_monomu_u_insert),s(t_h4s_nums_num,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar3),s(t_h4s_nums_num,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c31, axiom, ![V_t, V_l0, V_l, V_c0]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3),s(t_h4s_integers_int,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),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),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_monomu_u_insert),s(t_h4s_integers_int,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3),s(t_h4s_integers_int,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c28, axiom, ![V_t, V_l0, V_l]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),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),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_varlistu_u_insert),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c27, axiom, ![V_t, V_l0, V_l]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),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_nums_num),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_varlistu_u_insert),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t)))))).
fof(ah4s_canonicals_varlistu_u_mergeu_u_ok, axiom, ![TV_u_27a]: ![V_sr]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr)))) => ![V_vm, V_x, V_y]: s(TV_u_27a,h4s_canonicals_interpu_u_vl(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_y))))) = 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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,h4s_canonicals_interpu_u_vl(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_lists_list(t_h4s_quotes_index),V_x))))),s(TV_u_27a,h4s_canonicals_interpu_u_vl(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_lists_list(t_h4s_quotes_index),V_y))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c20, axiom, ![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_nilu_u_monom))) = 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_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_nilu_u_monom)))).
fof(ah4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_scalaru_u_def, axiom, s(t_fun(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_scalar) = s(t_fun(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_scalar(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_scalaru_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_c0]: 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(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_scalar(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),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_scalaru_u_defu_c0, axiom, ![TV_u_27a]: ![V_t, V_r, V_l, V_c0, V_c]: 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(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_scalar(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),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_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),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_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_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),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)),happ(s(t_fun(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_scalar(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_scalaru_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_r, V_l, V_c0]: 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(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_scalar(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),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_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),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_canonicals_consu_u_monom),s(TV_u_27a,V_c0))),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)),happ(s(t_fun(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_scalar(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_ringNorms_ru_u_varlistu_u_insertu_u_def, axiom, ![TV_u_27a]: ![V_r]: 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_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_canonicals_varlistu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_ringNorms_ru_u_canonicalu_u_sumu_u_scalar2u_u_def, axiom, ![TV_u_27a]: ![V_r]: 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_canonicalu_u_sumu_u_scalar2(s(t_h4s_rings_ring(TV_u_27a),V_r))) = 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_canonicals_canonicalu_u_sumu_u_scalar2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_numRings_numu_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_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_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_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_canonicals_varlistu_u_insert(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_numRings_numu_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_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_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_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_canonicals_canonicalu_u_sumu_u_scalar2(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_integerRings_intu_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_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_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_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_integerRings_intu_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_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_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_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3u_u_def, axiom, s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3) = s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_scalar3u_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_l0, V_c0]: 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_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_numRings_numu_u_canonicalu_u_sumu_u_scalar3u_u_def, axiom, s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar3) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)))),h4s_canonicals_canonicalu_u_sumu_u_scalar3(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_rings_semiu_u_ringu_u_ofu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,h4s_rings_ringu_u_r0(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_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(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)))))).
fof(ah4s_ringNorms_ru_u_canonicalu_u_sumu_u_scalar3u_u_def, axiom, ![TV_u_27a]: ![V_r]: 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_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(TV_u_27a),V_r))) = 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_canonicals_canonicalu_u_sumu_u_scalar3(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
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_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_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),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_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),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_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_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_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),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_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),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_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),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_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),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_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),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_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_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_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),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_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),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_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_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_bottom) => s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical0) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))))).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_bottom) => ![V_x, V_x0, V_x1]: 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_canonicals_u_20u_40indu_u_typecanonical1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),V_x0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x1))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_canonicals_u_20u_40indu_u_typecanonical4(s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))))))).
fof(ah4s_canonicals_Consu_u_varlist0, axiom, ![TV_u_27a]: 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_canonicals_consu_u_varlist) = 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_canonicals_u_20u_40indu_u_typecanonical2)).
fof(ah4s_canonicals_Consu_u_monom0, axiom, ![TV_u_27a]: 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_canonicals_consu_u_monom) = 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_canonicals_u_20u_40indu_u_typecanonical1)).
fof(ah4s_canonicals_Nilu_u_monom0, axiom, ![TV_u_27a]: 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_u_20u_40indu_u_typecanonical0)).
fof(ah4s_canonicals_canonicalu_u_sumu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27canonicalu_u_sumu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))) | (?[V_a0, V_a1, V_a2]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_lists_list(t_h4s_quotes_index),V_a1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),t_bool),V_uu_27canonicalu_u_sumu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a2))))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_lists_list(t_h4s_quotes_index),V_a0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),t_bool),V_uu_27canonicalu_u_sumu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a1))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),t_bool),V_uu_27canonicalu_u_sumu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),t_bool),V_uu_27canonicalu_u_sumu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_canonicals_u_20u_40indu_u_typecanonical4(s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),V_r)))).
fof(ah4s_canonicals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typecanonical2u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_bottom) => ![V_x, V_x0]: 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_canonicals_u_20u_40indu_u_typecanonical2),s(t_h4s_lists_list(t_h4s_quotes_index),V_x))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x0))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_lists_list(t_h4s_quotes_index),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_canonicals_u_20u_40indu_u_typecanonical4(s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index)))),V_uu_0))))))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_u_20u_40indu_u_typecanonical3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_h4s_quotes_index))),h4s_canonicals_u_20u_40indu_u_typecanonical4(s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_a))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_a)).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c25, axiom, ![V_t, V_l, V_c0]: 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_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_scalar),s(t_h4s_rats_rat,V_c0))),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_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),V_t))))) = 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_canonicals_consu_u_monom),s(t_h4s_rats_rat,V_c0))),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)),happ(s(t_fun(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_scalar),s(t_h4s_rats_rat,V_c0))),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),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_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),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_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_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_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),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_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),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_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_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),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_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),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_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),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_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),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_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),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_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),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_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),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_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_c17, axiom, ![V_v8, V_v7]: 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_nilu_u_monom))),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_canonicals_consu_u_varlist),s(t_h4s_lists_list(t_h4s_quotes_index),V_v7))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_v8))))) = 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_canonicals_consu_u_varlist),s(t_h4s_lists_list(t_h4s_quotes_index),V_v7))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_v8)))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c15, axiom, ![V_s1]: 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_s1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_s1)).
fof(ah4s_rats_RATu_u_SUBu_u_RDISTRIB, 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,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_a),s(t_h4s_rats_rat,V_b))))),s(t_h4s_rats_rat,V_c))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(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_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,V_b))),s(t_h4s_rats_rat,V_c)))))).
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,t0)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_canonicals_interpu_u_vlu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr]: s(TV_u_27a,h4s_canonicals_interpu_u_vl(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr)))).
fof(ah4s_canonicals_interpu_u_vlu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_vm, V_t, V_sr]: s(TV_u_27a,h4s_canonicals_interpu_u_vl(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),s(t_h4s_quotes_varmap(TV_u_27a),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(TV_u_27a,h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),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),V_t)))).
fof(ah4s_canonicals_ivlu_u_auxu_u_defu_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_vm, V_tu_27, V_sr]: s(TV_u_27a,h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),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,h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),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_canonicals_ivlu_u_auxu_u_defu_c0, axiom, ![TV_u_27a]: ![V_x, V_vm, V_sr]: s(TV_u_27a,h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr),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_quotes_indexu_u_ltu_u_def, axiom, ![V_i2, V_i1]: (p(s(t_bool,happ(s(t_fun(t_h4s_quotes_index,t_bool),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_index,V_i2)))) <=> s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_index,V_i2))) = s(t_h4s_prelims_ordering,h4s_prelims_less))).
fof(ah4s_semiu_u_rings_multu_u_oneu_u_right, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n]: 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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))))) = s(TV_u_27a,V_n))).
fof(ah4s_semiu_u_rings_multu_u_rotate, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_m, V_n, V_p]: 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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_n))))),s(TV_u_27a,V_p))) = 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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_p))))),s(TV_u_27a,V_m))))).
fof(ah4s_semiu_u_rings_multu_u_oneu_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n]: 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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n))).
fof(ah4s_prelims_listu_u_mergeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_l1, V_au_u_lt]: s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),V_l1)).
fof(ah4s_semiu_u_rings_multu_u_assoc, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n, V_m, V_p]: 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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))) = 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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),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_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_prelims_listu_u_mergeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v5, V_v4, V_au_u_lt]: s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_v4),s(t_h4s_lists_list(TV_u_27a),V_v5))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_v4),s(t_h4s_lists_list(TV_u_27a),V_v5)))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_prelims_listu_u_mergeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_y, V_x, V_l2, V_l1, V_au_u_lt]: s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_y),s(t_h4s_lists_list(TV_u_27a),V_l2))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_y),s(t_h4s_lists_list(TV_u_27a),V_l2))))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_y),s(t_h4s_lists_list(TV_u_27a),h4s_prelims_listu_u_merge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_au_u_lt),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2)))))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_rats_RATu_u_AINVu_u_LMUL, axiom, ![V_r2, V_r1]: 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_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,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),h4s_rats_ratu_u_ainv),s(t_h4s_rats_rat,V_r1))))),s(t_h4s_rats_rat,V_r2)))).
fof(ah4s_rats_RATu_u_SUBu_u_ADDAINV, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,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_r1))),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,V_r2)))))).
fof(ah4s_rats_RATu_u_RDISTRIB, 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,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_a))),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_add),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_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,V_b))),s(t_h4s_rats_rat,V_c)))))).
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(ch4s_ratRings_ratu_u_ringu_u_thmsu_c30, conjecture, ![V_t, V_l0, V_l, V_c0, V_c]: 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_canonicalu_u_sumu_u_scalar3),s(t_h4s_rats_rat,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),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_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),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,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_c0))),s(t_h4s_rats_rat,V_c))))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_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_canonicalu_u_sumu_u_scalar3),s(t_h4s_rats_rat,V_c0))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t)))))).
