%   ORIGINAL: h4/ringNorm/polynom__simplify__ok
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/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/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/semi__ring/semi__ring__accessors_c0: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/semi__ring/semi__ring__accessors_c1: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring0 a a0 f f0) = a0
% Assm: h4/semi__ring/semi__ring__accessors_c2: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% 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/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/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/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/interp__m__def_c0: !vm sr c. h4/canonical/interp__m sr vm c h4/list/NIL = c
% Assm: h4/canonical/interp__m__def_c1: !x vm t sr c. h4/canonical/interp__m sr vm c (h4/list/CONS x t) = h4/semi__ring/semi__ring__SRM sr c (h4/canonical/ivl__aux sr vm x t)
% Assm: h4/canonical/ics__aux__def_c0: !vm sr a. h4/canonical/ics__aux sr vm a h4/canonical/Nil__monom = a
% Assm: h4/canonical/ics__aux__def_c1: !vm t sr l a. h4/canonical/ics__aux sr vm a (h4/canonical/Cons__varlist l t) = h4/semi__ring/semi__ring__SRP sr a (h4/canonical/ics__aux sr vm (h4/canonical/interp__vl sr vm l) t)
% Assm: h4/canonical/ics__aux__def_c2: !vm t sr l c a. h4/canonical/ics__aux sr vm a (h4/canonical/Cons__monom c l t) = h4/semi__ring/semi__ring__SRP sr a (h4/canonical/ics__aux sr vm (h4/canonical/interp__m sr vm c l) t)
% Assm: h4/canonical/interp__cs__def_c0: !vm sr. h4/canonical/interp__cs sr vm h4/canonical/Nil__monom = h4/semi__ring/semi__ring__SR0 sr
% Assm: h4/canonical/interp__cs__def_c1: !vm t sr l. h4/canonical/interp__cs sr vm (h4/canonical/Cons__varlist l t) = h4/canonical/ics__aux sr vm (h4/canonical/interp__vl sr vm l) t
% Assm: h4/canonical/interp__cs__def_c2: !vm t sr l c. h4/canonical/interp__cs sr vm (h4/canonical/Cons__monom c l t) = h4/canonical/ics__aux sr vm (h4/canonical/interp__m sr vm c l) t
% Assm: h4/canonical/canonical__sum__merge__ok: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm x y. h4/canonical/interp__cs sr vm (h4/canonical/canonical__sum__merge sr x y) = h4/semi__ring/semi__ring__SRP sr (h4/canonical/interp__cs sr vm x) (h4/canonical/interp__cs sr vm y))
% Assm: h4/canonical/canonical__sum__scalar3__ok: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm c l s. h4/canonical/interp__cs sr vm (h4/canonical/canonical__sum__scalar3 sr c l s) = h4/semi__ring/semi__ring__SRM sr (h4/semi__ring/semi__ring__SRM sr c (h4/canonical/interp__vl sr vm l)) (h4/canonical/interp__cs sr vm s))
% Assm: h4/canonical/canonical__sum__prod__ok: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm x y. h4/canonical/interp__cs sr vm (h4/canonical/canonical__sum__prod sr x y) = h4/semi__ring/semi__ring__SRM sr (h4/canonical/interp__cs sr vm x) (h4/canonical/interp__cs sr vm y))
% Assm: h4/canonical/canonical__sum__simplify__ok: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm s. h4/canonical/interp__cs sr vm (h4/canonical/canonical__sum__simplify sr s) = h4/canonical/interp__cs sr vm s)
% Assm: h4/ring/mult__one__left: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RM r (h4/ring/ring__R1 r) n = n)
% 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/ring/ring__is__semi__ring: !r. h4/ring/is__ring r ==> h4/semi__ring/is__semi__ring (h4/ring/semi__ring__of r)
% Assm: h4/ring/neg__mult: !r. h4/ring/is__ring r ==> (!a b. h4/ring/ring__RM r (h4/ring/ring__RN r a) b = h4/ring/ring__RN r (h4/ring/ring__RM r a b))
% Assm: h4/ringNorm/r__interp__cs__def: !r. h4/ringNorm/r__interp__cs r = h4/canonical/interp__cs (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__ics__aux__def: !r. h4/ringNorm/r__ics__aux r = h4/canonical/ics__aux (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__interp__m__def: !r. h4/ringNorm/r__interp__m r = h4/canonical/interp__m (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__interp__vl__def: !r. h4/ringNorm/r__interp__vl r = h4/canonical/interp__vl (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__ivl__aux__def: !r. h4/ringNorm/r__ivl__aux r = h4/canonical/ivl__aux (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__canonical__sum__simplify__def: !r. h4/ringNorm/r__canonical__sum__simplify r = h4/canonical/canonical__sum__simplify (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__canonical__sum__prod__def: !r. h4/ringNorm/r__canonical__sum__prod r = h4/canonical/canonical__sum__prod (h4/ring/semi__ring__of 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/ringNorm/r__canonical__sum__merge__def: !r. h4/ringNorm/r__canonical__sum__merge r = h4/canonical/canonical__sum__merge (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/polynom__repfns_c0: !a. h4/ringNorm/_20_40ind__typeringNorm5 (h4/ringNorm/_20_40ind__typeringNorm6 a) = a
% Assm: h4/ringNorm/polynom__repfns_c1: !r. (\a0_27. !_27polynom_27. (!a0_270. (?a. a0_270 = (\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a) \/ (?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 h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27polynom_27 a0 /\ _27polynom_27 a1) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27polynom_27 a0 /\ _27polynom_27 a1) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) a /\ _27polynom_27 a) ==> _27polynom_27 a0_270) ==> _27polynom_27 a0_27) r <=> h4/ringNorm/_20_40ind__typeringNorm6 (h4/ringNorm/_20_40ind__typeringNorm5 r) = r
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm0____def: h4/ringNorm/_20_40ind__typeringNorm0 = (\a. h4/ringNorm/_20_40ind__typeringNorm5 ((\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm1____def: h4/ringNorm/_20_40ind__typeringNorm1 = (\a. h4/ringNorm/_20_40ind__typeringNorm5 ((\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm2____def: h4/ringNorm/_20_40ind__typeringNorm2 = (\a0 a1. h4/ringNorm/_20_40ind__typeringNorm5 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/ringNorm/_20_40ind__typeringNorm6 a0) (h4/ringNorm/_20_40ind__typeringNorm6 a1)))
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm3____def: h4/ringNorm/_20_40ind__typeringNorm3 = (\a0 a1. h4/ringNorm/_20_40ind__typeringNorm5 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/ringNorm/_20_40ind__typeringNorm6 a0) (h4/ringNorm/_20_40ind__typeringNorm6 a1)))
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm4____def: h4/ringNorm/_20_40ind__typeringNorm4 = (\a. h4/ringNorm/_20_40ind__typeringNorm5 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) (h4/ringNorm/_20_40ind__typeringNorm6 a)))
% Assm: h4/ringNorm/Pvar0: h4/ringNorm/Pvar = h4/ringNorm/_20_40ind__typeringNorm0
% Assm: h4/ringNorm/Pconst0: h4/ringNorm/Pconst = h4/ringNorm/_20_40ind__typeringNorm1
% Assm: h4/ringNorm/Pplus0: h4/ringNorm/Pplus = h4/ringNorm/_20_40ind__typeringNorm2
% Assm: h4/ringNorm/Pmult0: h4/ringNorm/Pmult = h4/ringNorm/_20_40ind__typeringNorm3
% Assm: h4/ringNorm/Popp0: h4/ringNorm/Popp = h4/ringNorm/_20_40ind__typeringNorm4
% Assm: h4/ringNorm/polynom__normalize__def_c0: !r i. h4/ringNorm/polynom__normalize r (h4/ringNorm/Pvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm: h4/ringNorm/polynom__normalize__def_c1: !r c. h4/ringNorm/polynom__normalize r (h4/ringNorm/Pconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm: h4/ringNorm/polynom__normalize__def_c2: !r pr pl. h4/ringNorm/polynom__normalize r (h4/ringNorm/Pplus pl pr) = h4/ringNorm/r__canonical__sum__merge r (h4/ringNorm/polynom__normalize r pl) (h4/ringNorm/polynom__normalize r pr)
% Assm: h4/ringNorm/polynom__normalize__def_c3: !r pr pl. h4/ringNorm/polynom__normalize r (h4/ringNorm/Pmult pl pr) = h4/ringNorm/r__canonical__sum__prod r (h4/ringNorm/polynom__normalize r pl) (h4/ringNorm/polynom__normalize r pr)
% Assm: h4/ringNorm/polynom__normalize__def_c4: !r p. h4/ringNorm/polynom__normalize r (h4/ringNorm/Popp p) = h4/ringNorm/r__canonical__sum__scalar3 r (h4/ring/ring__RN r (h4/ring/ring__R1 r)) h4/list/NIL (h4/ringNorm/polynom__normalize r p)
% Assm: h4/ringNorm/polynom__simplify__def: !x r. h4/ringNorm/polynom__simplify r x = h4/ringNorm/r__canonical__sum__simplify r (h4/ringNorm/polynom__normalize r x)
% Assm: h4/ringNorm/interp__p__def_c0: !vm r c. h4/ringNorm/interp__p r vm (h4/ringNorm/Pconst c) = c
% Assm: h4/ringNorm/interp__p__def_c1: !vm r i. h4/ringNorm/interp__p r vm (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm: h4/ringNorm/interp__p__def_c2: !vm r p2 p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Pplus p1 p2) = h4/ring/ring__RP r (h4/ringNorm/interp__p r vm p1) (h4/ringNorm/interp__p r vm p2)
% Assm: h4/ringNorm/interp__p__def_c3: !vm r p2 p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Pmult p1 p2) = h4/ring/ring__RM r (h4/ringNorm/interp__p r vm p1) (h4/ringNorm/interp__p r vm p2)
% Assm: h4/ringNorm/interp__p__def_c4: !vm r p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Popp p1) = h4/ring/ring__RN r (h4/ringNorm/interp__p r vm p1)
% Goal: !r. h4/ring/is__ring r ==> (!vm p. h4/ringNorm/r__interp__cs r vm (h4/ringNorm/polynom__simplify r p) = h4/ringNorm/interp__p r vm p)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_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_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c1]: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring0 a a0 f f0) = a0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% 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_canonicals_ivlu_u_auxu_u_defu_c0]: !x vm sr. happ (happ (happ (h4/canonical/ivl__aux sr) vm) x) h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_canonicals_ivlu_u_auxu_u_defu_c1]: !x_27 x vm t_27 sr. happ (happ (happ (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)) (happ (happ (happ (h4/canonical/ivl__aux sr) vm) x_27) t_27)
% Assm [h4s_canonicals_interpu_u_vlu_u_defu_c0]: !vm sr. happ (happ (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. happ (happ (h4/canonical/interp__vl sr) vm) (h4/list/CONS x t) = happ (happ (happ (h4/canonical/ivl__aux sr) vm) x) t
% Assm [h4s_canonicals_interpu_u_mu_u_defu_c0]: !vm sr c. happ (happ (happ (h4/canonical/interp__m sr) vm) c) h4/list/NIL = c
% Assm [h4s_canonicals_interpu_u_mu_u_defu_c1]: !x vm t sr c. happ (happ (happ (h4/canonical/interp__m sr) vm) c) (h4/list/CONS x t) = happ (happ (h4/semi__ring/semi__ring__SRM sr) c) (happ (happ (happ (h4/canonical/ivl__aux sr) vm) x) t)
% Assm [h4s_canonicals_icsu_u_auxu_u_defu_c0]: !vm sr a. happ (happ (happ (h4/canonical/ics__aux sr) vm) a) h4/canonical/Nil__monom = a
% Assm [h4s_canonicals_icsu_u_auxu_u_defu_c1]: !vm t sr l a. happ (happ (happ (h4/canonical/ics__aux sr) vm) a) (h4/canonical/Cons__varlist l t) = happ (happ (h4/semi__ring/semi__ring__SRP sr) a) (happ (happ (happ (h4/canonical/ics__aux sr) vm) (happ (happ (h4/canonical/interp__vl sr) vm) l)) t)
% Assm [h4s_canonicals_icsu_u_auxu_u_defu_c2]: !vm t sr l c a. happ (happ (happ (h4/canonical/ics__aux sr) vm) a) (h4/canonical/Cons__monom c l t) = happ (happ (h4/semi__ring/semi__ring__SRP sr) a) (happ (happ (happ (h4/canonical/ics__aux sr) vm) (happ (happ (happ (h4/canonical/interp__m sr) vm) c) l)) t)
% Assm [h4s_canonicals_interpu_u_csu_u_defu_c0]: !vm sr. happ (happ (h4/canonical/interp__cs sr) vm) h4/canonical/Nil__monom = h4/semi__ring/semi__ring__SR0 sr
% Assm [h4s_canonicals_interpu_u_csu_u_defu_c1]: !vm t sr l. happ (happ (h4/canonical/interp__cs sr) vm) (h4/canonical/Cons__varlist l t) = happ (happ (happ (h4/canonical/ics__aux sr) vm) (happ (happ (h4/canonical/interp__vl sr) vm) l)) t
% Assm [h4s_canonicals_interpu_u_csu_u_defu_c2]: !vm t sr l c. happ (happ (h4/canonical/interp__cs sr) vm) (h4/canonical/Cons__monom c l t) = happ (happ (happ (h4/canonical/ics__aux sr) vm) (happ (happ (happ (h4/canonical/interp__m sr) vm) c) l)) t
% Assm [h4s_canonicals_canonicalu_u_sumu_u_mergeu_u_ok]: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm x y. happ (happ (h4/canonical/interp__cs sr) vm) (happ (happ (h4/canonical/canonical__sum__merge sr) x) y) = happ (happ (h4/semi__ring/semi__ring__SRP sr) (happ (happ (h4/canonical/interp__cs sr) vm) x)) (happ (happ (h4/canonical/interp__cs sr) vm) y))
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalar3u_u_ok]: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm c l s. happ (happ (h4/canonical/interp__cs sr) vm) (happ (happ (happ (h4/canonical/canonical__sum__scalar3 sr) c) l) s) = happ (happ (h4/semi__ring/semi__ring__SRM sr) (happ (happ (h4/semi__ring/semi__ring__SRM sr) c) (happ (happ (h4/canonical/interp__vl sr) vm) l))) (happ (happ (h4/canonical/interp__cs sr) vm) s))
% Assm [h4s_canonicals_canonicalu_u_sumu_u_produ_u_ok]: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm x y. happ (happ (h4/canonical/interp__cs sr) vm) (happ (happ (h4/canonical/canonical__sum__prod sr) x) y) = happ (happ (h4/semi__ring/semi__ring__SRM sr) (happ (happ (h4/canonical/interp__cs sr) vm) x)) (happ (happ (h4/canonical/interp__cs sr) vm) y))
% Assm [h4s_canonicals_canonicalu_u_sumu_u_simplifyu_u_ok]: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm s. happ (happ (h4/canonical/interp__cs sr) vm) (happ (h4/canonical/canonical__sum__simplify sr) s) = happ (happ (h4/canonical/interp__cs sr) vm) s)
% Assm [h4s_rings_multu_u_oneu_u_left]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RM r) (h4/ring/ring__R1 r)) n = n)
% 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_rings_ringu_u_isu_u_semiu_u_ring]: !r. h4/ring/is__ring r ==> h4/semi__ring/is__semi__ring (h4/ring/semi__ring__of r)
% Assm [h4s_rings_negu_u_mult]: !r. h4/ring/is__ring r ==> (!a b. happ (happ (h4/ring/ring__RM r) (h4/ring/ring__RN r a)) b = h4/ring/ring__RN r (happ (happ (h4/ring/ring__RM r) a) b))
% Assm [h4s_ringNorms_ru_u_interpu_u_csu_u_def]: !r. h4/ringNorm/r__interp__cs r = h4/canonical/interp__cs (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_icsu_u_auxu_u_def]: !r. h4/ringNorm/r__ics__aux r = h4/canonical/ics__aux (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_interpu_u_mu_u_def]: !r. h4/ringNorm/r__interp__m r = h4/canonical/interp__m (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_interpu_u_vlu_u_def]: !r. h4/ringNorm/r__interp__vl r = h4/canonical/interp__vl (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_ivlu_u_auxu_u_def]: !r. h4/ringNorm/r__ivl__aux r = h4/canonical/ivl__aux (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_canonicalu_u_sumu_u_simplifyu_u_def]: !r. h4/ringNorm/r__canonical__sum__simplify r = h4/canonical/canonical__sum__simplify (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_canonicalu_u_sumu_u_produ_u_def]: !r. h4/ringNorm/r__canonical__sum__prod r = h4/canonical/canonical__sum__prod (h4/ring/semi__ring__of 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_ringNorms_ru_u_canonicalu_u_sumu_u_mergeu_u_def]: !r. h4/ringNorm/r__canonical__sum__merge r = h4/canonical/canonical__sum__merge (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_polynomu_u_repfnsu_c0]: !a. h4/ringNorm/_20_40ind__typeringNorm5 (h4/ringNorm/_20_40ind__typeringNorm6 a) = a
% Assm [h4s_ringNorms_polynomu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27polynom_27. (!a0_270. (?a. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a h4/bool/ARB) _0) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a) _0) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27polynom_27 a0 /\ happ _27polynom_27 a1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27polynom_27 a0 /\ happ _27polynom_27 a1) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a _0) /\ happ _27polynom_27 a) ==> happ _27polynom_27 a0_270) ==> happ _27polynom_27 r) <=> h4/ringNorm/_20_40ind__typeringNorm6 (h4/ringNorm/_20_40ind__typeringNorm5 r) = r)
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/ringNorm/_20_40ind__typeringNorm0 x = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x h4/bool/ARB) _0))
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/ringNorm/_20_40ind__typeringNorm1 x = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB x) _0))
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/ringNorm/_20_40ind__typeringNorm2 x) x' = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x') _0))))
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/ringNorm/_20_40ind__typeringNorm3 x) x' = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x') _0))))
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm4u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/ringNorm/_20_40ind__typeringNorm4 x = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x) _0)))
% Assm [h4s_ringNorms_Pvar0]: h4/ringNorm/Pvar = h4/ringNorm/_20_40ind__typeringNorm0
% Assm [h4s_ringNorms_Pconst0]: h4/ringNorm/Pconst = h4/ringNorm/_20_40ind__typeringNorm1
% Assm [h4s_ringNorms_Pplus0]: h4/ringNorm/Pplus = h4/ringNorm/_20_40ind__typeringNorm2
% Assm [h4s_ringNorms_Pmult0]: h4/ringNorm/Pmult = h4/ringNorm/_20_40ind__typeringNorm3
% Assm [h4s_ringNorms_Popp0]: h4/ringNorm/Popp = h4/ringNorm/_20_40ind__typeringNorm4
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c0]: !r i. h4/ringNorm/polynom__normalize r (happ h4/ringNorm/Pvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c1]: !r c. h4/ringNorm/polynom__normalize r (happ h4/ringNorm/Pconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c2]: !r pr pl. h4/ringNorm/polynom__normalize r (happ (happ h4/ringNorm/Pplus pl) pr) = happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/ringNorm/polynom__normalize r pl)) (h4/ringNorm/polynom__normalize r pr)
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c3]: !r pr pl. h4/ringNorm/polynom__normalize r (happ (happ h4/ringNorm/Pmult pl) pr) = happ (happ (h4/ringNorm/r__canonical__sum__prod r) (h4/ringNorm/polynom__normalize r pl)) (h4/ringNorm/polynom__normalize r pr)
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c4]: !r p. h4/ringNorm/polynom__normalize r (happ h4/ringNorm/Popp p) = happ (happ (happ (h4/ringNorm/r__canonical__sum__scalar3 r) (h4/ring/ring__RN r (h4/ring/ring__R1 r))) h4/list/NIL) (h4/ringNorm/polynom__normalize r p)
% Assm [h4s_ringNorms_polynomu_u_simplifyu_u_def]: !x r. h4/ringNorm/polynom__simplify r x = happ (h4/ringNorm/r__canonical__sum__simplify r) (h4/ringNorm/polynom__normalize r x)
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c0]: !vm r c. h4/ringNorm/interp__p r vm (happ h4/ringNorm/Pconst c) = c
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c1]: !vm r i. h4/ringNorm/interp__p r vm (happ h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c2]: !vm r p2 p1. h4/ringNorm/interp__p r vm (happ (happ h4/ringNorm/Pplus p1) p2) = happ (happ (h4/ring/ring__RP r) (h4/ringNorm/interp__p r vm p1)) (h4/ringNorm/interp__p r vm p2)
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c3]: !vm r p2 p1. h4/ringNorm/interp__p r vm (happ (happ h4/ringNorm/Pmult p1) p2) = happ (happ (h4/ring/ring__RM r) (h4/ringNorm/interp__p r vm p1)) (h4/ringNorm/interp__p r vm p2)
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c4]: !vm r p1. h4/ringNorm/interp__p r vm (happ h4/ringNorm/Popp p1) = h4/ring/ring__RN r (h4/ringNorm/interp__p r vm p1)
% Goal: !r. h4/ring/is__ring r ==> (!vm p. happ (happ (h4/ringNorm/r__interp__cs r) vm) (h4/ringNorm/polynom__simplify r p) = h4/ringNorm/interp__p r vm p)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q289964,TV_Q289960]: ![V_f, V_g]: (![V_x]: s(TV_Q289960,happ(s(t_fun(TV_Q289964,TV_Q289960),V_f),s(TV_Q289964,V_x))) = s(TV_Q289960,happ(s(t_fun(TV_Q289964,TV_Q289960),V_g),s(TV_Q289964,V_x))) => s(t_fun(TV_Q289964,TV_Q289960),V_f) = s(t_fun(TV_Q289964,TV_Q289960),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_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_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(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(TV_u_27a,V_a)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: 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),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(TV_u_27a,V_a0)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2, 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_srp(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_f)).
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_canonicals_ivlu_u_auxu_u_defu_c0, axiom, ![TV_u_27a]: ![V_x, V_vm, V_sr]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_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_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,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_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,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_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_interpu_u_vlu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_quotes_index),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,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_quotes_index),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,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_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_interpu_u_mu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(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(TV_u_27a,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(TV_u_27a,V_c)).
fof(ah4s_canonicals_interpu_u_mu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_vm, V_t, V_sr, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(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(TV_u_27a,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t))))) = s(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_c))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_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_icsu_u_auxu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_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(TV_u_27a,V_a))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(TV_u_27a,V_a)).
fof(ah4s_canonicals_icsu_u_auxu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_t, V_sr, V_l, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_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(TV_u_27a,V_a))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t))))) = s(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_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_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(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_quotes_index),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_l))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_canonicals_icsu_u_auxu_u_defu_c2, axiom, ![TV_u_27a]: ![V_vm, V_t, V_sr, V_l, V_c, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_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(TV_u_27a,V_a))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t))))) = s(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_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_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(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(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(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)))))).
fof(ah4s_canonicals_interpu_u_csu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr)))).
fof(ah4s_canonicals_interpu_u_csu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_t, V_sr, V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_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(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_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(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_quotes_index),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_l))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))).
fof(ah4s_canonicals_interpu_u_csu_u_defu_c2, axiom, ![TV_u_27a]: ![V_vm, V_t, V_sr, V_l, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_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(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_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(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(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(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)))).
fof(ah4s_canonicals_canonicalu_u_sumu_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,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_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_canonicals_canonicalu_u_sumu_u_merge(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_canonicals_canonicalu_u_sum(TV_u_27a),V_x))))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_canonicals_canonicalu_u_sum(TV_u_27a),V_y))))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalar3u_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_c, V_l, V_s]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_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_c))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_s))))) = 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,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_c))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_quotes_index),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_l))))))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_canonicals_canonicalu_u_sum(TV_u_27a),V_s))))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_produ_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,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_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_canonicals_canonicalu_u_sumu_u_prod(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_x))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_canonicals_canonicalu_u_sum(TV_u_27a),V_x))))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_canonicals_canonicalu_u_sum(TV_u_27a),V_y))))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_simplifyu_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_s]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_canonicals_canonicalu_u_sumu_u_simplify(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_s))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_canonicals_canonicalu_u_sum(TV_u_27a),V_s))))).
fof(ah4s_rings_multu_u_oneu_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_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_rings_ringu_u_rm(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_n))) = s(TV_u_27a,V_n))).
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_rings_ringu_u_isu_u_semiu_u_ring, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => 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),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))))).
fof(ah4s_rings_negu_u_mult, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_a, V_b]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_a))))),s(TV_u_27a,V_b))) = s(TV_u_27a,h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,happ(s(t_fun(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_a))),s(TV_u_27a,V_b))))))).
fof(ah4s_ringNorms_ru_u_interpu_u_csu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_cs(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(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_icsu_u_auxu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_ringnorms_ru_u_icsu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_u_aux(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_interpu_u_mu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_interpu_u_m(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(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_interpu_u_vlu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_vl(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),h4s_canonicals_interpu_u_vl(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_ivlu_u_auxu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_ivlu_u_aux(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_simplifyu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_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_simplify(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_produ_u_def, axiom, ![TV_u_27a]: ![V_r]: 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_prod(s(t_h4s_rings_ring(TV_u_27a),V_r))) = 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_canonicals_canonicalu_u_sumu_u_prod(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_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_ringNorms_ru_u_canonicalu_u_sumu_u_mergeu_u_def, axiom, ![TV_u_27a]: ![V_r]: 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_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_canonicals_canonicalu_u_sumu_u_merge(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_polynomu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_a))))) = s(t_h4s_ringNorms_polynom(TV_u_27a),V_a)).
fof(ah4s_ringNorms_polynomu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27polynomu_27]: (![V_a0u_270]: ((?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_a),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_a))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | ?[V_a]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r)))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm0),s(t_h4s_quotes_index,V_x))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_x),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm1),s(TV_u_27a,V_x))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm2u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_u_20u_40indu_u_typeringnorm2),s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm3u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_u_20u_40indu_u_typeringnorm3),s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm4u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm4),s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))).
fof(ah4s_ringNorms_Pvar0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar) = s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm0)).
fof(ah4s_ringNorms_Pconst0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst) = s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm1)).
fof(ah4s_ringNorms_Pplus0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus) = s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_u_20u_40indu_u_typeringnorm2)).
fof(ah4s_ringNorms_Pmult0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult) = s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_u_20u_40indu_u_typeringnorm3)).
fof(ah4s_ringNorms_Popp0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp) = s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm4)).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_r, V_i]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_i),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_r, V_c]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_c))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pl))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pr))))) = 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_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pl))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pr)))))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c3, axiom, ![TV_u_27a]: ![V_r, V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pl))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pr))))) = 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_prod(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pl))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pr)))))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c4, axiom, ![TV_u_27a]: ![V_r, V_p]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))) = 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,h4s_rings_ringu_u_rn(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),h4s_lists_nil))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p)))))).
fof(ah4s_ringNorms_polynomu_u_simplifyu_u_def, axiom, ![TV_u_27a]: ![V_x, V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),V_x)))))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_r, V_c]: s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_c))))) = s(TV_u_27a,V_c)).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_r, V_i]: s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c2, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p2, V_p1]: s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))),s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2)))))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c3, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p2, V_p1]: s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))),s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2)))))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c4, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p1]: s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))) = s(TV_u_27a,h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1)))))).
fof(ch4s_ringNorms_polynomu_u_simplifyu_u_ok, conjecture, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_vm, V_p]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_cs(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_ringnorms_polynomu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))) = s(TV_u_27a,h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r),s(t_h4s_quotes_varmap(TV_u_27a),V_vm),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))).
