%   ORIGINAL: h4/semi__ring/mult__zero__right
% 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/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/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/semi__ring/hidden____20__40ind____typesemi____ring0____def: h4/semi__ring/_20_40ind__typesemi__ring0 = (\a0 a1 a2 a3. h4/semi__ring/_20_40ind__typesemi__ring1 ((\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3))
% Assm: h4/semi__ring/semi__ring__repfns_c1: !r. (\a0_27. !_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3) ==> _27semi__ring_27 a0_270) ==> _27semi__ring_27 a0_27) r <=> h4/semi__ring/_20_40ind__typesemi__ring2 (h4/semi__ring/_20_40ind__typesemi__ring1 r) = r
% Assm: h4/semi__ring/semi__ring1: h4/semi__ring/semi__ring0 = h4/semi__ring/_20_40ind__typesemi__ring0
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/semi__ring/semi__ring__repfns_c0: !a. h4/semi__ring/_20_40ind__typesemi__ring1 (h4/semi__ring/_20_40ind__typesemi__ring2 a) = a
% Assm: h4/semi__ring/mult__zero__left: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SR0 r) n = h4/semi__ring/semi__ring__SR0 r)
% Assm: h4/semi__ring/plus__zero__right: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRP r n (h4/semi__ring/semi__ring__SR0 r) = n)
% Assm: h4/semi__ring/plus__zero__left: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SR0 r) n = n)
% Assm: h4/semi__ring/semi__ring__accessors_c0: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/semi__ring/semi__ring__SR00: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/semi__ring/semi__ring__component__equality: !s2 s1. s1 = s2 <=> h4/semi__ring/semi__ring__SR0 s1 = h4/semi__ring/semi__ring__SR0 s2 /\ h4/semi__ring/semi__ring__SR1 s1 = h4/semi__ring/semi__ring__SR1 s2 /\ h4/semi__ring/semi__ring__SRP s1 = h4/semi__ring/semi__ring__SRP s2 /\ h4/semi__ring/semi__ring__SRM s1 = h4/semi__ring/semi__ring__SRM s2
% Assm: h4/semi__ring/mult__one__right: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRM r n (h4/semi__ring/semi__ring__SR1 r) = n)
% Assm: h4/semi__ring/distr__left: !r. h4/semi__ring/is__semi__ring 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/semi__ring/mult__one__left: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SR1 r) n = n)
% Assm: h4/semi__ring/mult__sym: !r. h4/semi__ring/is__semi__ring r ==> (!n m. h4/semi__ring/semi__ring__SRM r n m = h4/semi__ring/semi__ring__SRM r m n)
% Assm: h4/semi__ring/mult__assoc: !r. h4/semi__ring/is__semi__ring r ==> (!n m p. h4/semi__ring/semi__ring__SRM r n (h4/semi__ring/semi__ring__SRM r m p) = h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r n m) p)
% Assm: h4/semi__ring/semi__ring__SRM0: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm: h4/semi__ring/semi__ring__accfupds_c12: !s f. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring__SR0__fupd f s) = f (h4/semi__ring/semi__ring__SR0 s)
% Assm: h4/semi__ring/semi__ring__accfupds_c2: !s f. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring__SRM__fupd f s) = h4/semi__ring/semi__ring__SR0 s
% Assm: h4/semi__ring/semi__ring__accfupds_c0: !s f. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring__SR1__fupd f s) = h4/semi__ring/semi__ring__SR0 s
% Assm: h4/semi__ring/semi__ring__accfupds_c1: !s f. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring__SRP__fupd f s) = h4/semi__ring/semi__ring__SR0 s
% 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/semi__ring/plus__sym: !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)
% Assm: h4/semi__ring/plus__assoc: !r. h4/semi__ring/is__semi__ring r ==> (!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)
% Assm: h4/semi__ring/semi__ring__fn__updates_c3: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SRM__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a a0 f (f1 f0)
% Assm: h4/semi__ring/semi__ring__fn__updates_c1: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SR1__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a (f1 a0) f f0
% Assm: h4/semi__ring/semi__ring__fn__updates_c0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SR0__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 (f1 a) a0 f f0
% Assm: h4/semi__ring/semi__ring__fn__updates_c2: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SRP__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a a0 (f1 f) f0
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/semi__ring/semi__ring__accfupds_c15: !s f. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring__SRM__fupd f s) = f (h4/semi__ring/semi__ring__SRM s)
% Assm: h4/semi__ring/semi__ring__accfupds_c9: !s f. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring__SR0__fupd f s) = h4/semi__ring/semi__ring__SRM s
% Assm: h4/semi__ring/semi__ring__accfupds_c11: !s f. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring__SRP__fupd f s) = h4/semi__ring/semi__ring__SRM s
% Assm: h4/semi__ring/semi__ring__accfupds_c10: !s f. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring__SR1__fupd f s) = h4/semi__ring/semi__ring__SRM s
% 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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/semi__ring/semi__ring__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3) ==> _27semi__ring_27 a0_270) ==> _27semi__ring_27 a0_27) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/semi__ring/FORALL__semi__ring: !P. (!s. P s) <=> (!a0 a f0 f. P (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0) (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a) (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0) (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f) h4/bool/ARB)))))
% Assm: h4/semi__ring/semi__ring__updates__eq__literal: !s f0 f a0 a. h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0) (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a) (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0) (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f) s))) = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0) (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a) (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0) (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f) h4/bool/ARB)))
% Assm: h4/semi__ring/semi__ring__literal__nchotomy: !s. ?a0 a f0 f. s = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0) (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a) (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0) (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f) h4/bool/ARB)))
% Assm: h4/semi__ring/EXISTS__semi__ring: !P. (?s. P s) <=> (?a0 a f0 f. P (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0) (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a) (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0) (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f) h4/bool/ARB)))))
% Assm: h4/semi__ring/semi__ring__case__cong: !f_27 f M_27 M. M = M_27 /\ (!a0 a1 a2 a3. M_27 = h4/semi__ring/semi__ring0 a0 a1 a2 a3 ==> f a0 a1 a2 a3 = f_27 a0 a1 a2 a3) ==> h4/semi__ring/semi__ring__CASE M f = h4/semi__ring/semi__ring__CASE M_27 f_27
% Assm: h4/semi__ring/semi__ring__nchotomy: !ss. ?a a0 f f0. ss = h4/semi__ring/semi__ring0 a a0 f f0
% Assm: h4/semi__ring/semi__ring__induction: !P. (!a a0 f f0. P (h4/semi__ring/semi__ring0 a a0 f f0)) ==> (!s. P s)
% Assm: h4/semi__ring/semi__ring__fupdfupds_c1: !s g f. h4/semi__ring/semi__ring__SR1__fupd f (h4/semi__ring/semi__ring__SR1__fupd g s) = h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g) s
% Assm: h4/semi__ring/semi__ring__fupdcanon_c4: !s g f. h4/semi__ring/semi__ring__SRM__fupd f (h4/semi__ring/semi__ring__SR1__fupd g s) = h4/semi__ring/semi__ring__SR1__fupd g (h4/semi__ring/semi__ring__SRM__fupd f s)
% Assm: h4/semi__ring/semi__ring__accfupds_c14: !s f. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring__SRP__fupd f s) = f (h4/semi__ring/semi__ring__SRP s)
% Assm: h4/semi__ring/semi__ring__accfupds_c8: !s f. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring__SRM__fupd f s) = h4/semi__ring/semi__ring__SRP s
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/semi__ring/semi__ring__accfupds_c5: !s f. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring__SRM__fupd f s) = h4/semi__ring/semi__ring__SR1 s
% Assm: h4/semi__ring/semi__ring__fupdcanon_c2: !s g f. h4/semi__ring/semi__ring__SRP__fupd f (h4/semi__ring/semi__ring__SR1__fupd g s) = h4/semi__ring/semi__ring__SR1__fupd g (h4/semi__ring/semi__ring__SRP__fupd f s)
% Assm: h4/semi__ring/semi__ring__accfupds_c7: !s f. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring__SR1__fupd f s) = h4/semi__ring/semi__ring__SRP s
% Assm: h4/semi__ring/semi__ring__accfupds_c4: !s f. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring__SRP__fupd f s) = h4/semi__ring/semi__ring__SR1 s
% Assm: h4/semi__ring/semi__ring__fupdfupds_c0: !s g f. h4/semi__ring/semi__ring__SR0__fupd f (h4/semi__ring/semi__ring__SR0__fupd g s) = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g) s
% Assm: h4/semi__ring/semi__ring__accfupds_c6: !s f. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring__SR0__fupd f s) = h4/semi__ring/semi__ring__SRP s
% Assm: h4/semi__ring/semi__ring__fupdcanon_c5: !s g f. h4/semi__ring/semi__ring__SRM__fupd f (h4/semi__ring/semi__ring__SRP__fupd g s) = h4/semi__ring/semi__ring__SRP__fupd g (h4/semi__ring/semi__ring__SRM__fupd f s)
% Assm: h4/semi__ring/semi__ring__fupdcanon_c1: !s g f. h4/semi__ring/semi__ring__SRP__fupd f (h4/semi__ring/semi__ring__SR0__fupd g s) = h4/semi__ring/semi__ring__SR0__fupd g (h4/semi__ring/semi__ring__SRP__fupd f s)
% Assm: h4/semi__ring/semi__ring__fupdcanon_c0: !s g f. h4/semi__ring/semi__ring__SR1__fupd f (h4/semi__ring/semi__ring__SR0__fupd g s) = h4/semi__ring/semi__ring__SR0__fupd g (h4/semi__ring/semi__ring__SR1__fupd f s)
% Assm: h4/semi__ring/semi__ring__accfupds_c13: !s f. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring__SR1__fupd f s) = f (h4/semi__ring/semi__ring__SR1 s)
% Assm: h4/semi__ring/semi__ring__accfupds_c3: !s f. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring__SR0__fupd f s) = h4/semi__ring/semi__ring__SR1 s
% Assm: h4/semi__ring/semi__ring__fupdcanon_c3: !s g f. h4/semi__ring/semi__ring__SRM__fupd f (h4/semi__ring/semi__ring__SR0__fupd g s) = h4/semi__ring/semi__ring__SR0__fupd g (h4/semi__ring/semi__ring__SRM__fupd f s)
% Assm: h4/semi__ring/semi__ring__fupdfupds_c2: !s g f. h4/semi__ring/semi__ring__SRP__fupd f (h4/semi__ring/semi__ring__SRP__fupd g s) = h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g) s
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/semi__ring/semi__ring__fupdfupds_c3: !s g f. h4/semi__ring/semi__ring__SRM__fupd f (h4/semi__ring/semi__ring__SRM__fupd g s) = h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g) s
% Assm: h4/semi__ring/semi__ring__case__def: !f a3 a2 a1 a0. h4/semi__ring/semi__ring__CASE (h4/semi__ring/semi__ring0 a0 a1 a2 a3) f = f a0 a1 a2 a3
% Assm: h4/semi__ring/semi__ring__SRM__fupd0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SRM__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a a0 f (f1 f0)
% Assm: h4/semi__ring/semi__ring__SR10: !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__SR0__fupd0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SR0__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 (f1 a) a0 f f0
% Assm: h4/semi__ring/semi__ring__SRP__fupd0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SRP__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a a0 (f1 f) f0
% Assm: h4/semi__ring/semi__ring__SR1__fupd0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SR1__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a (f1 a0) f f0
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/semi__ring/semi__ring__SRP0: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/combin/o__ASSOC: !h g f. h4/combin/o f (h4/combin/o g h) = h4/combin/o (h4/combin/o f g) h
% Assm: h4/semi__ring/semi__ring__11: !a3_27 a3 a2_27 a2 a1_27 a1 a0_27 a0. h4/semi__ring/semi__ring0 a0 a1 a2 a3 = h4/semi__ring/semi__ring0 a0_27 a1_27 a2_27 a3_27 <=> a0 = a0_27 /\ a1 = a1_27 /\ a2 = a2_27 /\ a3 = a3_27
% Assm: h4/semi__ring/semi__ring__size__def: !f a3 a2 a1 a0. h4/semi__ring/semi__ring__size f (h4/semi__ring/semi__ring0 a0 a1 a2 a3) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (f a1))
% Assm: h4/semi__ring/semi__ring__literal__11: !f2 f1 f02 f01 a2 a1 a02 a01. h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a01) (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a1) (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f01) (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f1) h4/bool/ARB))) = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a02) (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a2) (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f02) (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f2) h4/bool/ARB))) <=> a01 = a02 /\ a1 = a2 /\ f01 = f02 /\ f1 = f2
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c4: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/semi__ring/semi__ring__SRP__fupd f)
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c1: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) h)
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c9: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) h)
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c3: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) h)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c6: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SRM__fupd g) = h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c7: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)) h
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c11: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) h)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c1: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g)) h
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c3: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)) h
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c10: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SRP__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd g) (h4/semi__ring/semi__ring__SRM__fupd f)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c2: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c5: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) h)
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c6: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/semi__ring/semi__ring__SRM__fupd f)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c4: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/semi__ring/semi__ring__SRP__fupd g) = h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g)
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c0: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/semi__ring/semi__ring__SR1__fupd f)
% Assm: h4/semi__ring/semi__ring__Axiom: !f. ?fn. !a0 a1 a2 a3. fn (h4/semi__ring/semi__ring0 a0 a1 a2 a3) = f a0 a1 a2 a3
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c8: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/semi__ring/semi__ring__SRM__fupd f)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c0: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g)
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c7: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) h)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c5: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g)) h
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c2: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/semi__ring/semi__ring__SRP__fupd f)
% Assm: h4/list/FILTER__COND__REWRITE_c1: !l h P. P h ==> h4/list/FILTER P (h4/list/CONS h l) = h4/list/CONS h (h4/list/FILTER P l)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/list/FILTER0_c0: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm: h4/list/FILTER0_c1: !t h P. h4/list/FILTER P (h4/list/CONS h t) = h4/bool/COND (P h) (h4/list/CONS h (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/list/LAST__DEF: !t h. h4/list/LAST (h4/list/CONS h t) = h4/bool/COND (t = h4/list/NIL) h (h4/list/LAST t)
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Goal: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRM r n (h4/semi__ring/semi__ring__SR0 r) = h4/semi__ring/semi__ring__SR0 r)
%   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_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_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_semiu_u_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesemiu_u_u_u_ring0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x' x x'. happ (happ (happ (happ h4/semi__ring/_20_40ind__typesemi__ring0 x) x') x) x' = h4/semi__ring/_20_40ind__typesemi__ring1 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x (h4/pair/_2C x' (h4/pair/_2C x x'))) _0))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 a3))) _0) ==> happ _27semi__ring_27 a0_270) ==> happ _27semi__ring_27 r) <=> h4/semi__ring/_20_40ind__typesemi__ring2 (h4/semi__ring/_20_40ind__typesemi__ring1 r) = r)
% Assm [h4s_semiu_u_rings_semiu_u_ring1]: h4/semi__ring/semi__ring0 = h4/semi__ring/_20_40ind__typesemi__ring0
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c0]: !a. h4/semi__ring/_20_40ind__typesemi__ring1 (h4/semi__ring/_20_40ind__typesemi__ring2 a) = a
% Assm [h4s_semiu_u_rings_multu_u_zerou_u_left]: !r. h4/semi__ring/is__semi__ring r ==> (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) (h4/semi__ring/semi__ring__SR0 r)) n = h4/semi__ring/semi__ring__SR0 r)
% Assm [h4s_semiu_u_rings_plusu_u_zerou_u_right]: !r. h4/semi__ring/is__semi__ring r ==> (!n. happ (happ (h4/semi__ring/semi__ring__SRP r) n) (h4/semi__ring/semi__ring__SR0 r) = n)
% Assm [h4s_semiu_u_rings_plusu_u_zerou_u_left]: !r. h4/semi__ring/is__semi__ring r ==> (!n. happ (happ (h4/semi__ring/semi__ring__SRP r) (h4/semi__ring/semi__ring__SR0 r)) n = n)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SR00]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_componentu_u_equality]: !s2 s1. s1 = s2 <=> h4/semi__ring/semi__ring__SR0 s1 = h4/semi__ring/semi__ring__SR0 s2 /\ h4/semi__ring/semi__ring__SR1 s1 = h4/semi__ring/semi__ring__SR1 s2 /\ h4/semi__ring/semi__ring__SRP s1 = h4/semi__ring/semi__ring__SRP s2 /\ h4/semi__ring/semi__ring__SRM s1 = h4/semi__ring/semi__ring__SRM s2
% Assm [h4s_semiu_u_rings_multu_u_oneu_u_right]: !r. h4/semi__ring/is__semi__ring r ==> (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) n) (h4/semi__ring/semi__ring__SR1 r) = n)
% Assm [h4s_semiu_u_rings_distru_u_left]: !r. h4/semi__ring/is__semi__ring 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_semiu_u_rings_multu_u_oneu_u_left]: !r. h4/semi__ring/is__semi__ring r ==> (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) (h4/semi__ring/semi__ring__SR1 r)) n = n)
% Assm [h4s_semiu_u_rings_multu_u_sym]: !r. h4/semi__ring/is__semi__ring r ==> (!n m. happ (happ (h4/semi__ring/semi__ring__SRM r) n) m = happ (happ (h4/semi__ring/semi__ring__SRM r) m) n)
% Assm [h4s_semiu_u_rings_multu_u_assoc]: !r. h4/semi__ring/is__semi__ring r ==> (!n m p. happ (happ (h4/semi__ring/semi__ring__SRM r) n) (happ (happ (h4/semi__ring/semi__ring__SRM r) m) p) = happ (happ (h4/semi__ring/semi__ring__SRM r) (happ (happ (h4/semi__ring/semi__ring__SRM r) n) m)) p)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SRM0]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c12]: !s f. h4/semi__ring/semi__ring__SR0 (happ (h4/semi__ring/semi__ring__SR0__fupd f) s) = happ f (h4/semi__ring/semi__ring__SR0 s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c2]: !s f. h4/semi__ring/semi__ring__SR0 (happ (h4/semi__ring/semi__ring__SRM__fupd f) s) = h4/semi__ring/semi__ring__SR0 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c0]: !s f. h4/semi__ring/semi__ring__SR0 (happ (h4/semi__ring/semi__ring__SR1__fupd f) s) = h4/semi__ring/semi__ring__SR0 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c1]: !s f. h4/semi__ring/semi__ring__SR0 (happ (h4/semi__ring/semi__ring__SRP__fupd f) s) = h4/semi__ring/semi__ring__SR0 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f0
% Assm [h4s_semiu_u_rings_plusu_u_sym]: !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)
% Assm [h4s_semiu_u_rings_plusu_u_assoc]: !r. h4/semi__ring/is__semi__ring r ==> (!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)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c3]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SRM__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) (happ f1 f0)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c1]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SR1__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) (happ f1 a0)) f) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SR0__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 (happ f1 a)) a0) f) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c2]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SRP__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) (happ f1 f)) f0
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c15]: !s f. h4/semi__ring/semi__ring__SRM (happ (h4/semi__ring/semi__ring__SRM__fupd f) s) = happ f (h4/semi__ring/semi__ring__SRM s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c9]: !s f. h4/semi__ring/semi__ring__SRM (happ (h4/semi__ring/semi__ring__SR0__fupd f) s) = h4/semi__ring/semi__ring__SRM s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c11]: !s f. h4/semi__ring/semi__ring__SRM (happ (h4/semi__ring/semi__ring__SRP__fupd f) s) = h4/semi__ring/semi__ring__SRM s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c10]: !s f. h4/semi__ring/semi__ring__SRM (happ (h4/semi__ring/semi__ring__SR1__fupd f) s) = h4/semi__ring/semi__ring__SRM s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (happ (happ (happ (happ 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 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a0
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 a3))) _1) ==> happ _27semi__ring_27 a0_270) ==> happ _27semi__ring_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_semiu_u_rings_FORALLu_u_semiu_u_ring]: !P. (!s. happ P s) <=> (!a0 a f0 f. happ P (happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0)) (happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a)) (happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0)) (happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f)) h4/bool/ARB)))))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_updatesu_u_equ_u_literal]: !s f0 f a0 a. happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0)) (happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a)) (happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0)) (happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f)) s))) = happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0)) (happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a)) (happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0)) (happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f)) h4/bool/ARB)))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_literalu_u_nchotomy]: !s. ?a0 a f0 f. s = happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0)) (happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a)) (happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0)) (happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f)) h4/bool/ARB)))
% Assm [h4s_semiu_u_rings_EXISTSu_u_semiu_u_ring]: !P. (?s. happ P s) <=> (?a0 a f0 f. happ P (happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a0)) (happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a)) (happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f0)) (happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f)) h4/bool/ARB)))))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_caseu_u_cong]: !f_27 f M_27 M. M = M_27 /\ (!a0 a1 a2 a3. M_27 = happ (happ (happ (happ h4/semi__ring/semi__ring0 a0) a1) a2) a3 ==> happ (happ (happ (happ f a0) a1) a2) a3 = happ (happ (happ (happ f_27 a0) a1) a2) a3) ==> h4/semi__ring/semi__ring__CASE M f = h4/semi__ring/semi__ring__CASE M_27 f_27
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_nchotomy]: !ss. ?a a0 f f0. ss = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_induction]: !P. (!a a0 f f0. happ P (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0)) ==> (!s. happ P s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c1]: !s g f. happ (h4/semi__ring/semi__ring__SR1__fupd f) (happ (h4/semi__ring/semi__ring__SR1__fupd g) s) = happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)) s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c4]: !s g f. happ (h4/semi__ring/semi__ring__SRM__fupd f) (happ (h4/semi__ring/semi__ring__SR1__fupd g) s) = happ (h4/semi__ring/semi__ring__SR1__fupd g) (happ (h4/semi__ring/semi__ring__SRM__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c14]: !s f. h4/semi__ring/semi__ring__SRP (happ (h4/semi__ring/semi__ring__SRP__fupd f) s) = happ f (h4/semi__ring/semi__ring__SRP s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c8]: !s f. h4/semi__ring/semi__ring__SRP (happ (h4/semi__ring/semi__ring__SRM__fupd f) s) = h4/semi__ring/semi__ring__SRP s
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c5]: !s f. h4/semi__ring/semi__ring__SR1 (happ (h4/semi__ring/semi__ring__SRM__fupd f) s) = h4/semi__ring/semi__ring__SR1 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c2]: !s g f. happ (h4/semi__ring/semi__ring__SRP__fupd f) (happ (h4/semi__ring/semi__ring__SR1__fupd g) s) = happ (h4/semi__ring/semi__ring__SR1__fupd g) (happ (h4/semi__ring/semi__ring__SRP__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c7]: !s f. h4/semi__ring/semi__ring__SRP (happ (h4/semi__ring/semi__ring__SR1__fupd f) s) = h4/semi__ring/semi__ring__SRP s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c4]: !s f. h4/semi__ring/semi__ring__SR1 (happ (h4/semi__ring/semi__ring__SRP__fupd f) s) = h4/semi__ring/semi__ring__SR1 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c0]: !s g f. happ (h4/semi__ring/semi__ring__SR0__fupd f) (happ (h4/semi__ring/semi__ring__SR0__fupd g) s) = happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g)) s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c6]: !s f. h4/semi__ring/semi__ring__SRP (happ (h4/semi__ring/semi__ring__SR0__fupd f) s) = h4/semi__ring/semi__ring__SRP s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c5]: !s g f. happ (h4/semi__ring/semi__ring__SRM__fupd f) (happ (h4/semi__ring/semi__ring__SRP__fupd g) s) = happ (h4/semi__ring/semi__ring__SRP__fupd g) (happ (h4/semi__ring/semi__ring__SRM__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c1]: !s g f. happ (h4/semi__ring/semi__ring__SRP__fupd f) (happ (h4/semi__ring/semi__ring__SR0__fupd g) s) = happ (h4/semi__ring/semi__ring__SR0__fupd g) (happ (h4/semi__ring/semi__ring__SRP__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c0]: !s g f. happ (h4/semi__ring/semi__ring__SR1__fupd f) (happ (h4/semi__ring/semi__ring__SR0__fupd g) s) = happ (h4/semi__ring/semi__ring__SR0__fupd g) (happ (h4/semi__ring/semi__ring__SR1__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c13]: !s f. h4/semi__ring/semi__ring__SR1 (happ (h4/semi__ring/semi__ring__SR1__fupd f) s) = happ f (h4/semi__ring/semi__ring__SR1 s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c3]: !s f. h4/semi__ring/semi__ring__SR1 (happ (h4/semi__ring/semi__ring__SR0__fupd f) s) = h4/semi__ring/semi__ring__SR1 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c3]: !s g f. happ (h4/semi__ring/semi__ring__SRM__fupd f) (happ (h4/semi__ring/semi__ring__SR0__fupd g) s) = happ (h4/semi__ring/semi__ring__SR0__fupd g) (happ (h4/semi__ring/semi__ring__SRM__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c2]: !s g f. happ (h4/semi__ring/semi__ring__SRP__fupd f) (happ (h4/semi__ring/semi__ring__SRP__fupd g) s) = happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g)) s
% Assm [h4s_combins_Ku_u_THM]: !y x. happ (h4/combin/K x) y = x
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c3]: !s g f. happ (h4/semi__ring/semi__ring__SRM__fupd f) (happ (h4/semi__ring/semi__ring__SRM__fupd g) s) = happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)) s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_caseu_u_def]: !f a3 a2 a1 a0. h4/semi__ring/semi__ring__CASE (happ (happ (happ (happ h4/semi__ring/semi__ring0 a0) a1) a2) a3) f = happ (happ (happ (happ f a0) a1) a2) a3
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SRMu_u_fupd0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SRM__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) (happ f1 f0)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SR10]: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SR0u_u_fupd0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SR0__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 (happ f1 a)) a0) f) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SRPu_u_fupd0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SRP__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) (happ f1 f)) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SR1u_u_fupd0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SR1__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) (happ f1 a0)) f) f0
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SRP0]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_combins_ou_u_ASSOC]: !h g f. h4/combin/o f (h4/combin/o g h) = h4/combin/o (h4/combin/o f g) h
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_11]: !a3_27 a3 a2_27 a2 a1_27 a1 a0_27 a0. happ (happ (happ (happ h4/semi__ring/semi__ring0 a0) a1) a2) a3 = happ (happ (happ (happ h4/semi__ring/semi__ring0 a0_27) a1_27) a2_27) a3_27 <=> a0 = a0_27 /\ a1 = a1_27 /\ a2 = a2_27 /\ a3 = a3_27
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_sizeu_u_def]: !f a3 a2 a1 a0. h4/semi__ring/semi__ring__size f (happ (happ (happ (happ h4/semi__ring/semi__ring0 a0) a1) a2) a3) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (happ f a1))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_literalu_u_11]: !f2 f1 f02 f01 a2 a1 a02 a01. happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a01)) (happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a1)) (happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f01)) (happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f1)) h4/bool/ARB))) = happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/K a02)) (happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/K a2)) (happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/K f02)) (happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/K f2)) h4/bool/ARB))) <=> a01 = a02 /\ a1 = a2 /\ f01 = f02 /\ f1 = f2
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c4]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/semi__ring/semi__ring__SRP__fupd f)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c1]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) h)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c9]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) h)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c3]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) h)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c6]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SRM__fupd g) = h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c7]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)) h
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c11]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) h)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c1]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g)) h
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c3]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)) h
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c10]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SRP__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd g) (h4/semi__ring/semi__ring__SRM__fupd f)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c2]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c5]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) h)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c6]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/semi__ring/semi__ring__SRM__fupd f)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c4]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/semi__ring/semi__ring__SRP__fupd g) = h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c0]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/semi__ring/semi__ring__SR1__fupd f)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_Axiom]: !f. ?fn. !a0 a1 a2 a3. happ fn (happ (happ (happ (happ h4/semi__ring/semi__ring0 a0) a1) a2) a3) = happ (happ (happ (happ f a0) a1) a2) a3
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c8]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/semi__ring/semi__ring__SRM__fupd f)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c0]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c7]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) h)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c5]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g)) h
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c2]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd g) (h4/semi__ring/semi__ring__SRP__fupd f)
% Assm [h4s_lists_FILTERu_u_CONDu_u_REWRITEu_c1]: !l h P. happ P h ==> h4/list/FILTER P (happ (happ h4/list/CONS h) l) = happ (happ h4/list/CONS h) (h4/list/FILTER P l)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lists_FILTER0u_c0]: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_FILTER0u_c1]: !t h P. h4/list/FILTER P (happ (happ h4/list/CONS h) t) = h4/bool/COND (happ P h) (happ (happ h4/list/CONS h) (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_lists_LASTu_u_DEF]: !t h. ?v. (v <=> t = h4/list/NIL) /\ h4/list/LAST (happ (happ h4/list/CONS h) t) = h4/bool/COND v h (h4/list/LAST t)
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Goal: !r. h4/semi__ring/is__semi__ring r ==> (!n. happ (happ (h4/semi__ring/semi__ring__SRM r) n) (h4/semi__ring/semi__ring__SR0 r) = h4/semi__ring/semi__ring__SR0 r)
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_Q1090072,TV_Q1090068]: ![V_f, V_g]: (![V_x]: s(TV_Q1090068,happ(s(t_fun(TV_Q1090072,TV_Q1090068),V_f),s(TV_Q1090072,V_x))) = s(TV_Q1090068,happ(s(t_fun(TV_Q1090072,TV_Q1090068),V_g),s(TV_Q1090072,V_x))) => s(t_fun(TV_Q1090072,TV_Q1090068),V_f) = s(t_fun(TV_Q1090072,TV_Q1090068),V_g))).
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_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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_semiu_u_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesemiu_u_u_u_ring0u_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_fun(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_fun(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_fun(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]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring0),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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(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_fun(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_fun(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_fun(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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0))))))),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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))))))).
fof(ah4s_semiu_u_rings_semiu_u_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_fun(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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27semiu_u_ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3]: 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_fun(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_fun(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_fun(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_fun(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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))))),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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r)))).
fof(ah4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring0)).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_a))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_a)).
fof(ah4s_semiu_u_rings_multu_u_zerou_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_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))))).
fof(ah4s_semiu_u_rings_plusu_u_zerou_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_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,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))).
fof(ah4s_semiu_u_rings_plusu_u_zerou_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_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))).
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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SR00, 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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_componentu_u_equality, axiom, ![TV_u_27a]: ![V_s2, V_s1]: (s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s1) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s2) <=> (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_s1))) = 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_s2))) & (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_s1))) = 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_s2))) & (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_s1))) = 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_s2))) & 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_s1))) = 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_s2)))))))).
fof(ah4s_semiu_u_rings_multu_u_oneu_u_right, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))))) = s(TV_u_27a,V_n))).
fof(ah4s_semiu_u_rings_distru_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n, 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_semiu_u_rings_multu_u_oneu_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n))).
fof(ah4s_semiu_u_rings_multu_u_sym, 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_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))))).
fof(ah4s_semiu_u_rings_multu_u_assoc, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SRM0, 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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c12, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),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_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c2, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c0, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c1, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_semiu_u_rings_plusu_u_sym, 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))))).
fof(ah4s_semiu_u_rings_plusu_u_assoc, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_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))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c3, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c1, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f1),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)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f1),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)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c2, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c15, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),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)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c9, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c11, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c10, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_1),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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))) <=> ![V_uu_27semiu_u_ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3]: 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_fun(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_fun(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_fun(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_fun(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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))))),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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_1))) => 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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_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_fun(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),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_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_rep))))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> 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_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_semiu_u_rings_FORALLu_u_semiu_u_ring, axiom, ![TV_u_27a]: ![V_P]: (![V_s]: p(s(t_bool,happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_bool),V_P),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))) <=> ![V_a0, V_a, V_f0, V_f]: p(s(t_bool,happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_bool),V_P),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_bools_arb)))))))))))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_updatesu_u_equ_u_literal, axiom, ![TV_u_27a]: ![V_s, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_bools_arb)))))))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_literalu_u_nchotomy, axiom, ![TV_u_27a]: ![V_s]: ?[V_a0, V_a, V_f0, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_bools_arb)))))))))).
fof(ah4s_semiu_u_rings_EXISTSu_u_semiu_u_ring, axiom, ![TV_u_27a]: ![V_P]: (?[V_s]: p(s(t_bool,happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_bool),V_P),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))) <=> ?[V_a0, V_a, V_f0, V_f]: p(s(t_bool,happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_bool),V_P),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_bools_arb)))))))))))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b]: ![V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_M) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_Mu_27) & ![V_a0, V_a1, V_a2, V_a3]: (s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_Mu_27) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))) => s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b),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)),TV_u_27b)),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)),TV_u_27b))),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)),TV_u_27b)))),V_f),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))) = s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b),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)),TV_u_27b)),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)),TV_u_27b))),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)),TV_u_27b)))),V_fu_27),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))) => s(TV_u_27b,h4s_semiu_u_rings_semiu_u_ringu_u_case(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_M),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)),TV_u_27b)))),V_f))) = s(TV_u_27b,h4s_semiu_u_rings_semiu_u_ringu_u_case(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_Mu_27),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)),TV_u_27b)))),V_fu_27))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_nchotomy, axiom, ![TV_u_27a]: ![V_ss]: ?[V_a, V_a0, V_f, V_f0]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_ss) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_induction, axiom, ![TV_u_27a]: ![V_P]: (![V_a, V_a0, V_f, V_f0]: p(s(t_bool,happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_bool),V_P),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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)))))) => ![V_s]: p(s(t_bool,happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_bool),V_P),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c1, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c4, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c14, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),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)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c8, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
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_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c5, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c2, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c7, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c4, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c0, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c6, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c5, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c1, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c0, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c13, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),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_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c3, axiom, ![TV_u_27a]: ![V_s, V_f]: 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),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = 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_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c3, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c2, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c3, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_caseu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_a3, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_semiu_u_rings_semiu_u_ringu_u_case(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))),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)),TV_u_27b)))),V_f))) = s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b),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)),TV_u_27b)),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)),TV_u_27b))),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)),TV_u_27b)))),V_f),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SRMu_u_fupd0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SR10, 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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_SR0u_u_fupd0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f1),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)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SRPu_u_fupd0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SR1u_u_fupd0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f1),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)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & 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_P),s(TV_u_27a,V_a)))))).
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_semiu_u_rings_semiu_u_ringu_u_SRP0, 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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_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_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_ou_u_ASSOC, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27d,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27d,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(t_fun(TV_u_27d,TV_u_27c),V_h))))) = s(t_fun(TV_u_27d,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(t_fun(TV_u_27d,TV_u_27c),V_h)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_11, axiom, ![TV_u_27a]: ![V_a3u_27, V_a3, V_a2u_27, V_a2, V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a0u_27))),s(TV_u_27a,V_a1u_27))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2u_27))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & (s(TV_u_27a,V_a1) = s(TV_u_27a,V_a1u_27) & (s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2u_27) & s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3u_27)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_sizeu_u_def, axiom, ![TV_u_27a]: ![V_f, V_a3, V_a2, V_a1, V_a0]: s(t_h4s_nums_num,h4s_semiu_u_rings_semiu_u_ringu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(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_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a1)))))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_literalu_u_11, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f02, V_f01, V_a2, V_a1, V_a02, V_a01]: (s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a01))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a1))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f01))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f1))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_bools_arb))))))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a02))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a2))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f02))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_k(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f2))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_bools_arb))))))))) <=> (s(TV_u_27a,V_a01) = s(TV_u_27a,V_a02) & (s(TV_u_27a,V_a1) = s(TV_u_27a,V_a2) & (s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f01) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f02) & s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f1) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f2)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c4, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c9, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c6, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c7, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c11, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c10, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c2, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c5, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c6, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c4, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c0, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: ?[V_fn]: ![V_a0, V_a1, V_a2, V_a3]: s(TV_u_27b,happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_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_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))) = s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b),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)),TV_u_27b)),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)),TV_u_27b))),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)),TV_u_27b)))),V_f),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c8, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c0, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c7, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c5, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c2, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_lists_FILTERu_u_CONDu_u_REWRITEu_c1, axiom, ![TV_u_27a]: ![V_l, V_h, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lists_FILTER0u_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_FILTER0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_LASTu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_h]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27a),V_t) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)) & s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,V_h),s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(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(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
fof(ch4s_semiu_u_rings_multu_u_zerou_u_right, conjecture, ![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_sr0(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))))).
