%   ORIGINAL: h4/integerRing/int__is__ring
% 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/ring/is__ring__def: !r. h4/ring/is__ring r <=> (!n m. h4/ring/ring__RP r n m = h4/ring/ring__RP r m n) /\ (!n m p. h4/ring/ring__RP r n (h4/ring/ring__RP r m p) = h4/ring/ring__RP r (h4/ring/ring__RP r n m) p) /\ (!n m. h4/ring/ring__RM r n m = h4/ring/ring__RM r m n) /\ (!n m p. h4/ring/ring__RM r n (h4/ring/ring__RM r m p) = h4/ring/ring__RM r (h4/ring/ring__RM r n m) p) /\ (!n. h4/ring/ring__RP r (h4/ring/ring__R0 r) n = n) /\ (!n. h4/ring/ring__RM r (h4/ring/ring__R1 r) n = n) /\ (!n. h4/ring/ring__RP r n (h4/ring/ring__RN r n) = h4/ring/ring__R0 r) /\ (!n m p. h4/ring/ring__RM r (h4/ring/ring__RP r n m) p = h4/ring/ring__RP r (h4/ring/ring__RM r n p) (h4/ring/ring__RM r m p))
% Assm: h4/integer/int__10: h4/integer/int__1 = h4/integer/int__ABS h4/integer/tint__1
% Assm: h4/integer/INT__10: ~(h4/integer/int__1 = h4/integer/int__0)
% Assm: h4/integer/int__00: h4/integer/int__0 = h4/integer/int__ABS h4/integer/tint__0
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/integer/int__of__num1_c1: !n. h4/integer/int__of__num (h4/num/SUC n) = h4/integer/int__add (h4/integer/int__of__num n) h4/integer/int__1
% Assm: h4/integer/INT__1: h4/integer/int__1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/quotient/REP__ABS__RSP: !rep abs REL. h4/quotient/QUOTIENT REL abs rep ==> (!x1 x2. REL x1 x2 ==> REL x1 (rep (abs x2)))
% Assm: h4/quotient/EQUIV__REFL__SYM__TRANS: !R. (!x y. R x y <=> R x = R y) <=> (!x. R x x) /\ (!x y. R x y ==> R y x) /\ (!x y z. R x y /\ R y z ==> R x z)
% Assm: h4/integer/int__QUOTIENT: h4/quotient/QUOTIENT h4/integer/tint__eq h4/integer/int__ABS h4/integer/int__REP
% Assm: h4/quotient/EQUIV__def: !E. h4/quotient/EQUIV E <=> (!x y. E x y <=> E x = E y)
% Assm: h4/integer/TINT__EQ__EQUIV: !q p. h4/integer/tint__eq p q <=> h4/integer/tint__eq p = h4/integer/tint__eq q
% Assm: h4/quotient/EQUALS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> R (rep x) (rep y))
% Assm: h4/quotient/EQUALS__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 y1 y2. R x1 x2 /\ R y1 y2 ==> (R x1 y1 <=> R x2 y2))
% Assm: h4/integer/INT__LT__MUL: !y x. h4/integer/int__lt h4/integer/int__0 x /\ h4/integer/int__lt h4/integer/int__0 y ==> h4/integer/int__lt h4/integer/int__0 (h4/integer/int__mul x y)
% Assm: h4/integer/NUM__POSINT__EX: !t. ~h4/integer/int__lt t h4/integer/int__0 ==> (?n. t = h4/integer/int__of__num n)
% Assm: h4/integer/int__of__num1_c0: h4/integer/int__of__num h4/num/0 = h4/integer/int__0
% Assm: h4/integer/INT__0: h4/integer/int__0 = h4/integer/int__of__num h4/num/0
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/quotient/APPLY__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (f x) (g y)))
% Assm: h4/quotient/IDENTITY__QUOTIENT: h4/quotient/QUOTIENT $equals h4/combin/I h4/combin/I
% Assm: h4/quotient/FUN__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/integer/int__of__num0: !T1. h4/integer/int__of__num T1 = h4/integer/int__ABS (h4/integer/tint__of__num T1)
% Assm: h4/integer/TINT__10: ~h4/integer/tint__eq h4/integer/tint__1 h4/integer/tint__0
% Assm: h4/integer/tint__of__num0_c1: !n. h4/integer/tint__of__num (h4/num/SUC n) = h4/integer/tint__add (h4/integer/tint__of__num n) h4/integer/tint__1
% Assm: h4/integer/TINT__EQ__AP: !q p. p = q ==> h4/integer/tint__eq p q
% Assm: h4/integer/TINT__ADD__WELLDEF: !y2 y1 x2 x1. h4/integer/tint__eq x1 x2 /\ h4/integer/tint__eq y1 y2 ==> h4/integer/tint__eq (h4/integer/tint__add x1 y1) (h4/integer/tint__add x2 y2)
% Assm: h4/integer/tint__of__num0_c0: h4/integer/tint__of__num h4/num/0 = h4/integer/tint__0
% Assm: h4/integer/int__add0: !T2 T1. h4/integer/int__add T1 T2 = h4/integer/int__ABS (h4/integer/tint__add (h4/integer/int__REP T1) (h4/integer/int__REP T2))
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/quotient/EQUIV__RES__FORALL: !P E. h4/quotient/EQUIV E ==> (h4/bool/RES__FORALL (h4/quotient/respects E) P <=> $forall P)
% Assm: h4/quotient/RES__FORALL__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f g. h4/quotient/_3D_3D_3D_3E R $equals f g ==> (h4/bool/RES__FORALL (h4/quotient/respects R) f <=> h4/bool/RES__FORALL (h4/quotient/respects R) g))
% Assm: h4/quotient/FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/LAMBDA__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. (\x. f x) = h4/quotient/_2D_2D_3E rep1 abs2 (\x. rep2 (f (abs1 x)))))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/EQ__TRANS: !z y x. x = y /\ y = z ==> x = z
% Assm: h4/ring/ring1: h4/ring/ring0 = h4/ring/_20_40ind__typering0
% Assm: h4/integer/int__lt0: !T2 T1. h4/integer/int__lt T1 T2 <=> h4/integer/tint__lt (h4/integer/int__REP T1) (h4/integer/int__REP T2)
% Assm: h4/integer/TINT__LT__WELLDEF: !y2 y1 x2 x1. h4/integer/tint__eq x1 x2 /\ h4/integer/tint__eq y1 y2 ==> (h4/integer/tint__lt x1 y1 <=> h4/integer/tint__lt x2 y2)
% 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/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/mult__zero__left: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RM r (h4/ring/ring__R0 r) n = h4/ring/ring__R0 r)
% Assm: h4/ring/mult__zero__right: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RM r n (h4/ring/ring__R0 r) = h4/ring/ring__R0 r)
% Assm: h4/ring/plus__zero__right: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RP r n (h4/ring/ring__R0 r) = n)
% Assm: h4/ring/plus__zero__left: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RP r (h4/ring/ring__R0 r) n = n)
% Assm: h4/ring/mult__assoc: !r. h4/ring/is__ring r ==> (!n m p. h4/ring/ring__RM r n (h4/ring/ring__RM r m p) = h4/ring/ring__RM r (h4/ring/ring__RM r n m) p)
% Assm: h4/ring/mult__one__right: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RM r n (h4/ring/ring__R1 r) = n)
% Assm: h4/ring/mult__sym: !r. h4/ring/is__ring r ==> (!n m. h4/ring/ring__RM r n m = h4/ring/ring__RM r m n)
% Assm: h4/ring/plus__sym: !r. h4/ring/is__ring r ==> (!n m. h4/ring/ring__RP r n m = h4/ring/ring__RP r m n)
% Assm: h4/ring/plus__assoc: !r. h4/ring/is__ring r ==> (!n m p. h4/ring/ring__RP r n (h4/ring/ring__RP r m p) = h4/ring/ring__RP r (h4/ring/ring__RP r n m) p)
% Assm: h4/ringNorm/polynom__simplify__ok: !r. h4/ring/is__ring r ==> (!vm p. h4/ringNorm/r__interp__cs r vm (h4/ringNorm/polynom__simplify r p) = h4/ringNorm/interp__p r vm p)
% Assm: h4/ring/distr__left: !r. h4/ring/is__ring r ==> (!n m p. h4/ring/ring__RM r (h4/ring/ring__RP r n m) p = h4/ring/ring__RP r (h4/ring/ring__RM r n p) (h4/ring/ring__RM r m p))
% Assm: h4/ring/opp__def: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RP r n (h4/ring/ring__RN r n) = h4/ring/ring__R0 r)
% Assm: h4/ringNorm/polynom__normalize__ok: !r. h4/ring/is__ring r ==> (!vm p. h4/ringNorm/r__interp__cs r vm (h4/ringNorm/polynom__normalize r p) = h4/ringNorm/interp__p r vm p)
% 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/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/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_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_c0: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/integer/TINT__ADD__LID: !x. h4/integer/tint__eq (h4/integer/tint__add h4/integer/tint__0 x) x
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/semi__ring/is__semi__ring__def: !r. h4/semi__ring/is__semi__ring r <=> (!n m. h4/semi__ring/semi__ring__SRP r n m = h4/semi__ring/semi__ring__SRP r m n) /\ (!n m p. h4/semi__ring/semi__ring__SRP r n (h4/semi__ring/semi__ring__SRP r m p) = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r n m) p) /\ (!n m. h4/semi__ring/semi__ring__SRM r n m = h4/semi__ring/semi__ring__SRM r m n) /\ (!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) /\ (!n. h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SR0 r) n = n) /\ (!n. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SR1 r) n = n) /\ (!n. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SR0 r) n = h4/semi__ring/semi__ring__SR0 r) /\ (!n m p. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRP r n m) p = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRM r n p) (h4/semi__ring/semi__ring__SRM r m p))
% Assm: h4/integer/int__mul0: !T2 T1. h4/integer/int__mul T1 T2 = h4/integer/int__ABS (h4/integer/tint__mul (h4/integer/int__REP T1) (h4/integer/int__REP T2))
% Assm: h4/integer/TINT__MUL__WELLDEF: !y2 y1 x2 x1. h4/integer/tint__eq x1 x2 /\ h4/integer/tint__eq y1 y2 ==> h4/integer/tint__eq (h4/integer/tint__mul x1 y1) (h4/integer/tint__mul x2 y2)
% Assm: h4/integer/TINT__LT__MUL: !y x. h4/integer/tint__lt h4/integer/tint__0 x /\ h4/integer/tint__lt h4/integer/tint__0 y ==> h4/integer/tint__lt h4/integer/tint__0 (h4/integer/tint__mul x y)
% Assm: h4/integer/NUM__POSTINT__EX: !t. ~h4/integer/tint__lt t h4/integer/tint__0 ==> (?n. h4/integer/tint__eq t (h4/integer/tint__of__num n))
% Assm: h4/ring/ring__induction: !P. (!a a0 f f0 f1. P (h4/ring/ring0 a a0 f f0 f1)) ==> (!r. P r)
% Assm: h4/ring/ring__repfns_c1: !r. (\a0_27. !_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = (\a00 a10 a20 a30 a40. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 (h4/pair/_2C a30 a40)))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3 a4) ==> _27ring_27 a0_270) ==> _27ring_27 a0_27) r <=> h4/ring/_20_40ind__typering2 (h4/ring/_20_40ind__typering1 r) = r
% Assm: h4/ring/hidden____20__40ind____typering0____def: h4/ring/_20_40ind__typering0 = (\a0 a1 a2 a3 a4. h4/ring/_20_40ind__typering1 ((\a00 a10 a20 a30 a40. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 (h4/pair/_2C a30 a40)))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3 a4))
% Assm: h4/ring/ring__repfns_c0: !a. h4/ring/_20_40ind__typering1 (h4/ring/_20_40ind__typering2 a) = a
% Assm: h4/semi__ring/plus__permute: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r m n) p = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r m p) n)
% 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/ringNorm/interp__p__def_c0: !vm r c. h4/ringNorm/interp__p r vm (h4/ringNorm/Pconst c) = c
% 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__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/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/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_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/Pmult0: h4/ringNorm/Pmult = h4/ringNorm/_20_40ind__typeringNorm3
% Assm: h4/ringNorm/Popp0: h4/ringNorm/Popp = h4/ringNorm/_20_40ind__typeringNorm4
% Assm: h4/ringNorm/Pplus0: h4/ringNorm/Pplus = h4/ringNorm/_20_40ind__typeringNorm2
% 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/Pconst0: h4/ringNorm/Pconst = h4/ringNorm/_20_40ind__typeringNorm1
% Assm: h4/ringNorm/Pvar0: h4/ringNorm/Pvar = h4/ringNorm/_20_40ind__typeringNorm0
% 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____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/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____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____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/polynom__repfns_c0: !a. h4/ringNorm/_20_40ind__typeringNorm5 (h4/ringNorm/_20_40ind__typeringNorm6 a) = a
% 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__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__ivl__aux__def: !r. h4/ringNorm/r__ivl__aux r = h4/canonical/ivl__aux (h4/ring/semi__ring__of r)
% 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/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/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/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__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__cs__def: !r. h4/ringNorm/r__interp__cs r = h4/canonical/interp__cs (h4/ring/semi__ring__of r)
% 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__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/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_c4: !vm r p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Popp p1) = h4/ring/ring__RN r (h4/ringNorm/interp__p r vm p1)
% Assm: h4/ringNorm/r__interp__m__def: !r. h4/ringNorm/r__interp__m r = h4/canonical/interp__m (h4/ring/semi__ring__of r)
% 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/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__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__vl__def_c0: !vm sr. h4/canonical/interp__vl sr vm h4/list/NIL = h4/semi__ring/semi__ring__SR1 sr
% 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_c0: !vm sr a. h4/canonical/ics__aux sr vm a h4/canonical/Nil__monom = a
% 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/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/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/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% 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/r__canonical__sum__simplify__def: !r. h4/ringNorm/r__canonical__sum__simplify r = h4/canonical/canonical__sum__simplify (h4/ring/semi__ring__of r)
% Goal: h4/ring/is__ring (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
%   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_rings_isu_u_ringu_u_def]: !r. h4/ring/is__ring r <=> (!n m. happ (happ (h4/ring/ring__RP r) n) m = happ (happ (h4/ring/ring__RP r) m) n) /\ (!n m p. happ (happ (h4/ring/ring__RP r) n) (happ (happ (h4/ring/ring__RP r) m) p) = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ring/ring__RP r) n) m)) p) /\ (!n m. happ (happ (h4/ring/ring__RM r) n) m = happ (happ (h4/ring/ring__RM r) m) n) /\ (!n m p. happ (happ (h4/ring/ring__RM r) n) (happ (happ (h4/ring/ring__RM r) m) p) = happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ring/ring__RM r) n) m)) p) /\ (!n. happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R0 r)) n = n) /\ (!n. happ (happ (h4/ring/ring__RM r) (h4/ring/ring__R1 r)) n = n) /\ (!n. happ (happ (h4/ring/ring__RP r) n) (h4/ring/ring__RN r n) = h4/ring/ring__R0 r) /\ (!n m p. happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ring/ring__RP r) n) m)) p = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ring/ring__RM r) n) p)) (happ (happ (h4/ring/ring__RM r) m) p))
% Assm [h4s_integers_intu_u_10]: h4/integer/int__1 = happ h4/integer/int__ABS h4/integer/tint__1
% Assm [h4s_integers_INTu_u_10]: ~(h4/integer/int__1 = h4/integer/int__0)
% Assm [h4s_integers_intu_u_00]: h4/integer/int__0 = happ h4/integer/int__ABS h4/integer/tint__0
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_integers_intu_u_ofu_u_num1u_c1]: !n. h4/integer/int__of__num (h4/num/SUC n) = happ (happ h4/integer/int__add (h4/integer/int__of__num n)) h4/integer/int__1
% Assm [h4s_integers_INTu_u_1]: h4/integer/int__1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_quotients_REPu_u_ABSu_u_RSP]: !rep abs REL. h4/quotient/QUOTIENT REL abs rep ==> (!x1 x2. happ (happ REL x1) x2 ==> happ (happ REL x1) (happ rep (happ abs x2)))
% Assm [h4s_quotients_EQUIVu_u_REFLu_u_SYMu_u_TRANS]: !R. (!x y. happ (happ R x) y <=> happ R x = happ R y) <=> (!x. happ (happ R x) x) /\ (!x y. happ (happ R x) y ==> happ (happ R y) x) /\ (!x y z. happ (happ R x) y /\ happ (happ R y) z ==> happ (happ R x) z)
% Assm [h4s_integers_intu_u_QUOTIENT]: h4/quotient/QUOTIENT h4/integer/tint__eq h4/integer/int__ABS h4/integer/int__REP
% Assm [h4s_quotients_EQUIVu_u_def]: !E. h4/quotient/EQUIV E <=> (!x y. happ (happ E x) y <=> happ E x = happ E y)
% Assm [h4s_integers_TINTu_u_EQu_u_EQUIV]: !q p. happ (happ h4/integer/tint__eq p) q <=> happ h4/integer/tint__eq p = happ h4/integer/tint__eq q
% Assm [h4s_quotients_EQUALSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> happ (happ R (happ rep x)) (happ rep y))
% Assm [h4s_quotients_EQUALSu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 y1 y2. happ (happ R x1) x2 /\ happ (happ R y1) y2 ==> (happ (happ R x1) y1 <=> happ (happ R x2) y2))
% Assm [h4s_integers_INTu_u_LTu_u_MUL]: !y x. h4/integer/int__lt h4/integer/int__0 x /\ h4/integer/int__lt h4/integer/int__0 y ==> h4/integer/int__lt h4/integer/int__0 (happ (happ h4/integer/int__mul x) y)
% Assm [h4s_integers_NUMu_u_POSINTu_u_EX]: !t. ~h4/integer/int__lt t h4/integer/int__0 ==> (?n. t = h4/integer/int__of__num n)
% Assm [h4s_integers_intu_u_ofu_u_num1u_c0]: h4/integer/int__of__num h4/num/0 = h4/integer/int__0
% Assm [h4s_integers_INTu_u_0]: h4/integer/int__0 = h4/integer/int__of__num h4/num/0
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_quotients_APPLYu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))
% Assm [h4s_quotients_IDENTITYu_u_QUOTIENT]: h4/quotient/QUOTIENT $equals h4/combin/I h4/combin/I
% Assm [h4s_quotients_FUNu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_integers_intu_u_ofu_u_num0]: !T1. h4/integer/int__of__num T1 = happ h4/integer/int__ABS (h4/integer/tint__of__num T1)
% Assm [h4s_integers_TINTu_u_10]: ~happ (happ h4/integer/tint__eq h4/integer/tint__1) h4/integer/tint__0
% Assm [h4s_integers_tintu_u_ofu_u_num0u_c1]: !n. h4/integer/tint__of__num (h4/num/SUC n) = h4/integer/tint__add (h4/integer/tint__of__num n) h4/integer/tint__1
% Assm [h4s_integers_TINTu_u_EQu_u_AP]: !q p. p = q ==> happ (happ h4/integer/tint__eq p) q
% Assm [h4s_integers_TINTu_u_ADDu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/integer/tint__eq x1) x2 /\ happ (happ h4/integer/tint__eq y1) y2 ==> happ (happ h4/integer/tint__eq (h4/integer/tint__add x1 y1)) (h4/integer/tint__add x2 y2)
% Assm [h4s_integers_tintu_u_ofu_u_num0u_c0]: h4/integer/tint__of__num h4/num/0 = h4/integer/tint__0
% Assm [h4s_integers_intu_u_add0]: !T2 T1. happ (happ h4/integer/int__add T1) T2 = happ h4/integer/int__ABS (h4/integer/tint__add (happ h4/integer/int__REP T1) (happ h4/integer/int__REP T2))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_quotients_EQUIVu_u_RESu_u_FORALL]: !P E. h4/quotient/EQUIV E ==> (h4/bool/RES__FORALL (h4/quotient/respects E) P <=> $forall P)
% Assm [h4s_quotients_RESu_u_FORALLu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f g. happ (happ (h4/quotient/_3D_3D_3D_3E R $equals) f) g ==> (h4/bool/RES__FORALL (h4/quotient/respects R) f <=> h4/bool/RES__FORALL (h4/quotient/respects R) g))
% Assm [h4s_quotients_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_LAMBDAu_u_PRS]: !_0. (!rep2 f abs1 x. happ (happ (happ (happ _0 rep2) f) abs1) x = happ rep2 (happ f (happ abs1 x))) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ (happ (h4/quotient/_2D_2D_3E rep1 abs2) (happ (happ (happ _0 rep2) f) abs1)) x)))
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_bools_EQu_u_TRANS]: !z y x. x = y /\ y = z ==> x = z
% Assm [h4s_rings_ring1]: h4/ring/ring0 = h4/ring/_20_40ind__typering0
% Assm [h4s_integers_intu_u_lt0]: !T2 T1. h4/integer/int__lt T1 T2 <=> h4/integer/tint__lt (happ h4/integer/int__REP T1) (happ h4/integer/int__REP T2)
% Assm [h4s_integers_TINTu_u_LTu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/integer/tint__eq x1) x2 /\ happ (happ h4/integer/tint__eq y1) y2 ==> (h4/integer/tint__lt x1 y1 <=> h4/integer/tint__lt x2 y2)
% 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_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_multu_u_zerou_u_left]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RM r) (h4/ring/ring__R0 r)) n = h4/ring/ring__R0 r)
% Assm [h4s_rings_multu_u_zerou_u_right]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RM r) n) (h4/ring/ring__R0 r) = h4/ring/ring__R0 r)
% Assm [h4s_rings_plusu_u_zerou_u_right]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RP r) n) (h4/ring/ring__R0 r) = n)
% Assm [h4s_rings_plusu_u_zerou_u_left]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R0 r)) n = n)
% Assm [h4s_rings_multu_u_assoc]: !r. h4/ring/is__ring r ==> (!n m p. happ (happ (h4/ring/ring__RM r) n) (happ (happ (h4/ring/ring__RM r) m) p) = happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ring/ring__RM r) n) m)) p)
% Assm [h4s_rings_multu_u_oneu_u_right]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RM r) n) (h4/ring/ring__R1 r) = n)
% Assm [h4s_rings_multu_u_sym]: !r. h4/ring/is__ring r ==> (!n m. happ (happ (h4/ring/ring__RM r) n) m = happ (happ (h4/ring/ring__RM r) m) n)
% Assm [h4s_rings_plusu_u_sym]: !r. h4/ring/is__ring r ==> (!n m. happ (happ (h4/ring/ring__RP r) n) m = happ (happ (h4/ring/ring__RP r) m) n)
% Assm [h4s_rings_plusu_u_assoc]: !r. h4/ring/is__ring r ==> (!n m p. happ (happ (h4/ring/ring__RP r) n) (happ (happ (h4/ring/ring__RP r) m) p) = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ring/ring__RP r) n) m)) p)
% Assm [h4s_ringNorms_polynomu_u_simplifyu_u_ok]: !r. h4/ring/is__ring r ==> (!vm p. happ (happ (h4/ringNorm/r__interp__cs r) vm) (h4/ringNorm/polynom__simplify r p) = h4/ringNorm/interp__p r vm p)
% Assm [h4s_rings_distru_u_left]: !r. h4/ring/is__ring r ==> (!n m p. happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ring/ring__RP r) n) m)) p = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ring/ring__RM r) n) p)) (happ (happ (h4/ring/ring__RM r) m) p))
% Assm [h4s_rings_oppu_u_def]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RP r) n) (h4/ring/ring__RN r n) = h4/ring/ring__R0 r)
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_ok]: !r. h4/ring/is__ring r ==> (!vm p. happ (happ (h4/ringNorm/r__interp__cs r) vm) (h4/ringNorm/polynom__normalize r p) = h4/ringNorm/interp__p r vm p)
% 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_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_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_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_c0]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm [h4s_integers_TINTu_u_ADDu_u_LID]: !x. happ (happ h4/integer/tint__eq (h4/integer/tint__add h4/integer/tint__0 x)) x
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_semiu_u_rings_isu_u_semiu_u_ringu_u_def]: !r. h4/semi__ring/is__semi__ring r <=> (!n m. happ (happ (h4/semi__ring/semi__ring__SRP r) n) m = happ (happ (h4/semi__ring/semi__ring__SRP r) m) n) /\ (!n m p. happ (happ (h4/semi__ring/semi__ring__SRP r) n) (happ (happ (h4/semi__ring/semi__ring__SRP r) m) p) = happ (happ (h4/semi__ring/semi__ring__SRP r) (happ (happ (h4/semi__ring/semi__ring__SRP r) n) m)) p) /\ (!n m. happ (happ (h4/semi__ring/semi__ring__SRM r) n) m = happ (happ (h4/semi__ring/semi__ring__SRM r) m) n) /\ (!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) /\ (!n. happ (happ (h4/semi__ring/semi__ring__SRP r) (h4/semi__ring/semi__ring__SR0 r)) n = n) /\ (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) (h4/semi__ring/semi__ring__SR1 r)) n = n) /\ (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) (h4/semi__ring/semi__ring__SR0 r)) n = h4/semi__ring/semi__ring__SR0 r) /\ (!n m p. happ (happ (h4/semi__ring/semi__ring__SRM r) (happ (happ (h4/semi__ring/semi__ring__SRP r) n) m)) p = happ (happ (h4/semi__ring/semi__ring__SRP r) (happ (happ (h4/semi__ring/semi__ring__SRM r) n) p)) (happ (happ (h4/semi__ring/semi__ring__SRM r) m) p))
% Assm [h4s_integers_intu_u_mul0]: !T2 T1. happ (happ h4/integer/int__mul T1) T2 = happ h4/integer/int__ABS (h4/integer/tint__mul (happ h4/integer/int__REP T1) (happ h4/integer/int__REP T2))
% Assm [h4s_integers_TINTu_u_MULu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/integer/tint__eq x1) x2 /\ happ (happ h4/integer/tint__eq y1) y2 ==> happ (happ h4/integer/tint__eq (h4/integer/tint__mul x1 y1)) (h4/integer/tint__mul x2 y2)
% Assm [h4s_integers_TINTu_u_LTu_u_MUL]: !y x. h4/integer/tint__lt h4/integer/tint__0 x /\ h4/integer/tint__lt h4/integer/tint__0 y ==> h4/integer/tint__lt h4/integer/tint__0 (h4/integer/tint__mul x y)
% Assm [h4s_integers_NUMu_u_POSTINTu_u_EX]: !t. ~h4/integer/tint__lt t h4/integer/tint__0 ==> (?n. happ (happ h4/integer/tint__eq t) (h4/integer/tint__of__num n))
% Assm [h4s_rings_ringu_u_induction]: !P. (!a a0 f f0 f1. happ P (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1)) ==> (!r. happ P r)
% Assm [h4s_rings_ringu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 (h4/pair/_2C a3 a4)))) _0) ==> happ _27ring_27 a0_270) ==> happ _27ring_27 r) <=> h4/ring/_20_40ind__typering2 (h4/ring/_20_40ind__typering1 r) = r)
% Assm [h4s_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typering0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x' x x' x. happ (happ (happ (happ (happ h4/ring/_20_40ind__typering0 x) x') x) x') x = h4/ring/_20_40ind__typering1 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x (h4/pair/_2C x' (h4/pair/_2C x (h4/pair/_2C x' x)))) _0))
% Assm [h4s_rings_ringu_u_repfnsu_c0]: !a. h4/ring/_20_40ind__typering1 (h4/ring/_20_40ind__typering2 a) = a
% Assm [h4s_semiu_u_rings_plusu_u_permute]: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. happ (happ (h4/semi__ring/semi__ring__SRP r) (happ (happ (h4/semi__ring/semi__ring__SRP r) m) n)) p = happ (happ (h4/semi__ring/semi__ring__SRP r) (happ (happ (h4/semi__ring/semi__ring__SRP r) m) p)) n)
% 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_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_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_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_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_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_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_Pmult0]: h4/ringNorm/Pmult = h4/ringNorm/_20_40ind__typeringNorm3
% Assm [h4s_ringNorms_Popp0]: h4/ringNorm/Popp = h4/ringNorm/_20_40ind__typeringNorm4
% Assm [h4s_ringNorms_Pplus0]: h4/ringNorm/Pplus = h4/ringNorm/_20_40ind__typeringNorm2
% 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_Pconst0]: h4/ringNorm/Pconst = h4/ringNorm/_20_40ind__typeringNorm1
% Assm [h4s_ringNorms_Pvar0]: h4/ringNorm/Pvar = h4/ringNorm/_20_40ind__typeringNorm0
% 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_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_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_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_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_polynomu_u_repfnsu_c0]: !a. h4/ringNorm/_20_40ind__typeringNorm5 (h4/ringNorm/_20_40ind__typeringNorm6 a) = a
% 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_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_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_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_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_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_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_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_csu_u_def]: !r. h4/ringNorm/r__interp__cs r = h4/canonical/interp__cs (h4/ring/semi__ring__of r)
% 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_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_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_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)
% 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_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_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_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_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_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_c0]: !vm sr a. happ (happ (happ (h4/canonical/ics__aux sr) vm) a) h4/canonical/Nil__monom = a
% 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_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_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_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_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% 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_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)
% Goal: h4/ring/is__ring (happ (happ (happ (happ (happ h4/ring/ring0 h4/integer/int__0) h4/integer/int__1) h4/integer/int__add) h4/integer/int__mul) h4/integer/int__neg)
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_Q1253956,TV_Q1253952]: ![V_f, V_g]: (![V_x]: s(TV_Q1253952,happ(s(t_fun(TV_Q1253956,TV_Q1253952),V_f),s(TV_Q1253956,V_x))) = s(TV_Q1253952,happ(s(t_fun(TV_Q1253956,TV_Q1253952),V_g),s(TV_Q1253956,V_x))) => s(t_fun(TV_Q1253956,TV_Q1253952),V_f) = s(t_fun(TV_Q1253956,TV_Q1253952),V_g))).
fof(ah4s_rings_isu_u_ringu_u_def, 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, V_m]: 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_n))),s(TV_u_27a,V_m))) = 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_m))),s(TV_u_27a,V_n))) & (![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_rings_ringu_u_rp(s(t_h4s_rings_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_rings_ringu_u_rp(s(t_h4s_rings_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_rings_ringu_u_rp(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_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) & (![V_n, V_m]: 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_n))),s(TV_u_27a,V_m))) = 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_m))),s(TV_u_27a,V_n))) & (![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_rings_ringu_u_rm(s(t_h4s_rings_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_rings_ringu_u_rm(s(t_h4s_rings_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_rings_ringu_u_rm(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_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) & (![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_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n) & (![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) & (![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_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_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_rings_ringu_u_rm(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_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),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_rings_ringu_u_rp(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_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_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p)))))))))))))).
fof(ah4s_integers_intu_u_10, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_1) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_1)))).
fof(ah4s_integers_INTu_u_10, axiom, ~ (s(t_h4s_integers_int,h4s_integers_intu_u_1) = s(t_h4s_integers_int,h4s_integers_intu_u_0))).
fof(ah4s_integers_intu_u_00, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_0) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0)))).
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_integers_intu_u_ofu_u_num1u_c1, axiom, ![V_n]: s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = 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_add),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))),s(t_h4s_integers_int,h4s_integers_intu_u_1)))).
fof(ah4s_integers_INTu_u_1, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_1) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_quotients_REPu_u_ABSu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_REL]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(TV_u_27a,V_x1))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_x2))))))))))).
fof(ah4s_quotients_EQUIVu_u_REFLu_u_SYMu_u_TRANS, axiom, ![TV_u_27a]: ![V_R]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y)))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))))) & ![V_x, V_y, V_z]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_z))))))))).
fof(ah4s_integers_intu_u_QUOTIENT, axiom, p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep))))).
fof(ah4s_quotients_EQUIVu_u_def, axiom, ![TV_u_27a]: ![V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_y)))))).
fof(ah4s_integers_TINTu_u_EQu_u_EQUIV, axiom, ![V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))))).
fof(ah4s_quotients_EQUALSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x, V_y]: (s(TV_u_27b,V_x) = s(TV_u_27b,V_y) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_y))))))))).
fof(ah4s_quotients_EQUALSu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2, V_y1, V_y2]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y1))),s(TV_u_27a,V_y2))))) => s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x1))),s(TV_u_27a,V_y1))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x2))),s(TV_u_27a,V_y2)))))).
fof(ah4s_integers_INTu_u_LTu_u_MUL, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,V_x)))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,V_y))))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_0),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_x))),s(t_h4s_integers_int,V_y)))))))).
fof(ah4s_integers_NUMu_u_POSINTu_u_EX, axiom, ![V_t]: (~ (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_t),s(t_h4s_integers_int,h4s_integers_intu_u_0))))) => ?[V_n]: s(t_h4s_integers_int,V_t) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))).
fof(ah4s_integers_intu_u_ofu_u_num1u_c0, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_integers_int,h4s_integers_intu_u_0)).
fof(ah4s_integers_INTu_u_0, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_0) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))).
fof(ah4s_quotients_APPLYu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_quotients_IDENTITYu_u_QUOTIENT, axiom, ![TV_u_27a]: p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))))).
fof(ah4s_quotients_FUNu_u_QUOTIENT, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_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_integers_intu_u_ofu_u_num0, axiom, ![V_T1]: s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_T1))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,V_T1)))))).
fof(ah4s_integers_TINTu_u_10, axiom, ~ (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0)))))).
fof(ah4s_integers_tintu_u_ofu_u_num0u_c1, axiom, ![V_n]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_1)))).
fof(ah4s_integers_TINTu_u_EQu_u_AP, axiom, ![V_q, V_p]: (s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))))).
fof(ah4s_integers_TINTu_u_ADDu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2)))))))).
fof(ah4s_integers_tintu_u_ofu_u_num0u_c0, axiom, s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0)).
fof(ah4s_integers_intu_u_add0, axiom, ![V_T2, V_T1]: 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_add),s(t_h4s_integers_int,V_T1))),s(t_h4s_integers_int,V_T2))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,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_quotients_EQUIVu_u_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_P, V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) => s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E))),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),V_P))))).
fof(ah4s_quotients_RESu_u_FORALLu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))),s(t_fun(TV_u_27a,t_bool),V_f))),s(t_fun(TV_u_27a,t_bool),V_g)))) => s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_g)))))).
fof(ah4s_quotients_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_forall(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_LAMBDAu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_rep2, V_f, V_abs1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_x))))))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))))),s(TV_u_27c,V_x))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_EQu_u_TRANS, axiom, ![TV_u_27a]: ![V_z, 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_z)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_z))).
fof(ah4s_rings_ring1, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering0)).
fof(ah4s_integers_intu_u_lt0, axiom, ![V_T2, V_T1]: s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_T1),s(t_h4s_integers_int,V_T2))) = s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T2)))))).
fof(ah4s_integers_TINTu_u_LTu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))) => s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))) = s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))).
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_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_multu_u_zerou_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_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))).
fof(ah4s_rings_multu_u_zerou_u_right, 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,V_n))),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_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))).
fof(ah4s_rings_plusu_u_zerou_u_right, 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_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,V_n))).
fof(ah4s_rings_plusu_u_zerou_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_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r0(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_multu_u_assoc, 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, 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_rings_ringu_u_rm(s(t_h4s_rings_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_rings_ringu_u_rm(s(t_h4s_rings_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_rings_ringu_u_rm(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_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))))).
fof(ah4s_rings_multu_u_oneu_u_right, 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,V_n))),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))).
fof(ah4s_rings_multu_u_sym, 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, V_m]: 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_n))),s(TV_u_27a,V_m))) = 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_m))),s(TV_u_27a,V_n))))).
fof(ah4s_rings_plusu_u_sym, 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, V_m]: 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_n))),s(TV_u_27a,V_m))) = 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_m))),s(TV_u_27a,V_n))))).
fof(ah4s_rings_plusu_u_assoc, 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, 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_rings_ringu_u_rp(s(t_h4s_rings_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_rings_ringu_u_rp(s(t_h4s_rings_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_rings_ringu_u_rp(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_rp(s(t_h4s_rings_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_ringNorms_polynomu_u_simplifyu_u_ok, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_vm, V_p]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_cs(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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))))).
fof(ah4s_rings_distru_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, 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_rings_ringu_u_rm(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_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),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_rings_ringu_u_rp(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_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_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))))).
fof(ah4s_rings_oppu_u_def, 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_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_ok, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_vm, V_p]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_cs(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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))))) = 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))))).
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_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_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_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_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_integers_TINTu_u_ADDu_u_LID, axiom, ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))))).
fof(ah4s_arithmetics_ONE, axiom, 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_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_semiu_u_rings_isu_u_semiu_u_ringu_u_def, 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]: 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_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))) = 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_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_n))) & (![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_srp(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_srp(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_srp(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_srp(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))) & (![V_n, V_m]: 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,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))) & (![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))) & (![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_srp(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_sr0(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) & (![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) & (![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_sr0(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_sr0(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,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_r))),s(TV_u_27a,V_n))),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_srp(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,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)))))))))))))).
fof(ah4s_integers_intu_u_mul0, axiom, ![V_T2, V_T1]: 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_T1))),s(t_h4s_integers_int,V_T2))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T2)))))))).
fof(ah4s_integers_TINTu_u_MULu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2)))))))).
fof(ah4s_integers_TINTu_u_LTu_u_MUL, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x)))) & p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))))) => p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y)))))))).
fof(ah4s_integers_NUMu_u_POSTINTu_u_EX, axiom, ![V_t]: (~ (p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_t),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0))))) => ?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_t))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_rings_ringu_u_induction, axiom, ![TV_u_27a]: ![V_P]: (![V_a, V_a0, V_f, V_f0, V_f1]: p(s(t_bool,happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_bool),V_P),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),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)))))) => ![V_r]: p(s(t_bool,happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_bool),V_P),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_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_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),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_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3, V_a4]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3),s(t_fun(TV_u_27a,TV_u_27a),V_a4))))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),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_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),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_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering2(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r)))).
fof(ah4s_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typering0u_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_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),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_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_x, V_xi_, V_x0, V_xi_0, V_x1]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering0),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0))),s(t_fun(TV_u_27a,TV_u_27a),V_x1))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_xi_),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0),s(t_fun(TV_u_27a,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_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))))))).
fof(ah4s_rings_ringu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering2(s(t_h4s_rings_ring(TV_u_27a),V_a))))) = s(t_h4s_rings_ring(TV_u_27a),V_a)).
fof(ah4s_semiu_u_rings_plusu_u_permute, 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_srp(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_srp(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_srp(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_srp(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,V_n))))).
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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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(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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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(ch4s_integerRings_intu_u_isu_u_ring, conjecture, p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(t_h4s_integers_int),happ(s(t_fun(t_fun(t_h4s_integers_int,t_h4s_integers_int),t_h4s_rings_ring(t_h4s_integers_int)),happ(s(t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),t_fun(t_fun(t_h4s_integers_int,t_h4s_integers_int),t_h4s_rings_ring(t_h4s_integers_int))),happ(s(t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),t_fun(t_fun(t_h4s_integers_int,t_h4s_integers_int),t_h4s_rings_ring(t_h4s_integers_int)))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),t_fun(t_fun(t_h4s_integers_int,t_h4s_integers_int),t_h4s_rings_ring(t_h4s_integers_int))))),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),t_fun(t_fun(t_h4s_integers_int,t_h4s_integers_int),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))))))).
