%   ORIGINAL: h4/ring/ring__is__semi__ring
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/ring/is__ring__def: !r. h4/ring/is__ring r <=> (!n m. h4/ring/ring__RP r n m = h4/ring/ring__RP r m n) /\ (!n m p. h4/ring/ring__RP r n (h4/ring/ring__RP r m p) = h4/ring/ring__RP r (h4/ring/ring__RP r n m) p) /\ (!n m. h4/ring/ring__RM r n m = h4/ring/ring__RM r m n) /\ (!n m p. h4/ring/ring__RM r n (h4/ring/ring__RM r m p) = h4/ring/ring__RM r (h4/ring/ring__RM r n m) p) /\ (!n. h4/ring/ring__RP r (h4/ring/ring__R0 r) n = n) /\ (!n. h4/ring/ring__RM r (h4/ring/ring__R1 r) n = n) /\ (!n. h4/ring/ring__RP r n (h4/ring/ring__RN r n) = h4/ring/ring__R0 r) /\ (!n m p. h4/ring/ring__RM r (h4/ring/ring__RP r n m) p = h4/ring/ring__RP r (h4/ring/ring__RM r n p) (h4/ring/ring__RM r m p))
% Assm: h4/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/ring/semi__ring__of__def: !r. h4/ring/semi__ring__of r = h4/semi__ring/semi__ring0 (h4/ring/ring__R0 r) (h4/ring/ring__R1 r) (h4/ring/ring__RP r) (h4/ring/ring__RM r)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__TRANS: !z y x. x = y /\ y = z ==> x = z
% Assm: h4/ring/plus__assoc: !r. h4/ring/is__ring r ==> (!n m p. h4/ring/ring__RP r n (h4/ring/ring__RP r m p) = h4/ring/ring__RP r (h4/ring/ring__RP r n m) p)
% Assm: h4/ring/mult__zero__left: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RM r (h4/ring/ring__R0 r) n = h4/ring/ring__R0 r)
% Assm: h4/ring/plus__sym: !r. h4/ring/is__ring r ==> (!n m. h4/ring/ring__RP r n m = h4/ring/ring__RP r m n)
% Assm: h4/ring/mult__assoc: !r. h4/ring/is__ring r ==> (!n m p. h4/ring/ring__RM r n (h4/ring/ring__RM r m p) = h4/ring/ring__RM r (h4/ring/ring__RM r n m) p)
% Assm: h4/ring/plus__zero__left: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RP r (h4/ring/ring__R0 r) n = n)
% Assm: h4/ring/mult__sym: !r. h4/ring/is__ring r ==> (!n m. h4/ring/ring__RM r n m = h4/ring/ring__RM r m n)
% Assm: h4/ring/opp__def: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RP r n (h4/ring/ring__RN r n) = h4/ring/ring__R0 r)
% Assm: h4/ring/mult__one__left: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RM r (h4/ring/ring__R1 r) n = n)
% Assm: h4/ring/plus__zero__right: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RP r n (h4/ring/ring__R0 r) = n)
% Assm: h4/ring/mult__zero__right: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RM r n (h4/ring/ring__R0 r) = h4/ring/ring__R0 r)
% Assm: h4/ring/distr__left: !r. h4/ring/is__ring r ==> (!n m p. h4/ring/ring__RM r (h4/ring/ring__RP r n m) p = h4/ring/ring__RP r (h4/ring/ring__RM r n p) (h4/ring/ring__RM r m p))
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/ring/ring__repfns_c0: !a. h4/ring/_20_40ind__typering1 (h4/ring/_20_40ind__typering2 a) = a
% Assm: h4/ring/ring1: h4/ring/ring0 = h4/ring/_20_40ind__typering0
% Assm: h4/ring/ring__repfns_c1: !r. (\a0_27. !_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = (\a00 a10 a20 a30 a40. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 (h4/pair/_2C a30 a40)))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3 a4) ==> _27ring_27 a0_270) ==> _27ring_27 a0_27) r <=> h4/ring/_20_40ind__typering2 (h4/ring/_20_40ind__typering1 r) = r
% Assm: h4/ring/hidden____20__40ind____typering0____def: h4/ring/_20_40ind__typering0 = (\a0 a1 a2 a3 a4. h4/ring/_20_40ind__typering1 ((\a00 a10 a20 a30 a40. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 (h4/pair/_2C a30 a40)))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3 a4))
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% 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/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/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/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/mult__rotate: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r m n) p = h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r n p) m)
% Assm: h4/semi__ring/mult__permute: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r m n) p = h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r m p) n)
% 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/mult__zero__right: !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)
% 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/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/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__permute: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r m n) p = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r m p) n)
% Assm: h4/semi__ring/distr__right: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRM r m (h4/semi__ring/semi__ring__SRP r n p) = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRM r m n) (h4/semi__ring/semi__ring__SRM r m p))
% Assm: h4/semi__ring/plus__rotate: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r m n) p = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r n p) m)
% Assm: h4/ring/ring__fn__updates_c1: !f2 f1 f0 f a0 a. h4/ring/ring__R1__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a (f2 a0) f f0 f1
% Assm: h4/ring/ring__fn__updates_c3: !f2 f1 f0 f a0 a. h4/ring/ring__RM__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 f (f2 f0) f1
% Assm: h4/ring/ring__fn__updates_c0: !f2 f1 f0 f a0 a. h4/ring/ring__R0__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 (f2 a) a0 f f0 f1
% Assm: h4/ring/ring__fn__updates_c4: !f2 f1 f0 f a0 a. h4/ring/ring__RN__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 f f0 (f2 f1)
% Assm: h4/ring/ring__fn__updates_c2: !f2 f1 f0 f a0 a. h4/ring/ring__RP__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 (f2 f) f0 f1
% Assm: h4/ring/ring__accessors_c2: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/ring/ring__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = (\a00 a10 a20 a30 a40. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 (h4/pair/_2C a30 a40)))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3 a4) ==> _27ring_27 a0_270) ==> _27ring_27 a0_27) 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/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/ring/ring__accessors_c4: !f1 f0 f a0 a. h4/ring/ring__RN (h4/ring/ring0 a a0 f f0 f1) = f1
% Assm: h4/ring/ring__accessors_c3: !f1 f0 f a0 a. h4/ring/ring__RM (h4/ring/ring0 a a0 f f0 f1) = f0
% Assm: h4/ring/ring__accfupds_c19: !r f. h4/ring/ring__RN (h4/ring/ring__RM__fupd f r) = h4/ring/ring__RN r
% Assm: h4/ring/ring__accfupds_c10: !r f. h4/ring/ring__RP (h4/ring/ring__RM__fupd f r) = h4/ring/ring__RP r
% Assm: h4/ring/ring__accfupds_c13: !r f. h4/ring/ring__RM (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RM r
% Assm: h4/ring/ring__induction: !P. (!a a0 f f0 f1. P (h4/ring/ring0 a a0 f f0 f1)) ==> (!r. P r)
% Assm: h4/ring/ring__accfupds_c12: !r f. h4/ring/ring__RM (h4/ring/ring__R0__fupd f r) = h4/ring/ring__RM r
% Assm: h4/ring/ring__fupdfupds_c4: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RN__fupd g r) = h4/ring/ring__RN__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__fupdcanon_c7: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RN__fupd f r)
% Assm: h4/ring/ring__fupdcanon_c9: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RM__fupd g r) = h4/ring/ring__RM__fupd g (h4/ring/ring__RN__fupd f r)
% Assm: h4/ring/ring__accfupds_c15: !r f. h4/ring/ring__RM (h4/ring/ring__RN__fupd f r) = h4/ring/ring__RM r
% Assm: h4/ring/ring__accfupds_c24: !r f. h4/ring/ring__RN (h4/ring/ring__RN__fupd f r) = f (h4/ring/ring__RN r)
% Assm: h4/ring/ring__accessors_c1: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm: h4/ring/ring__accessors_c0: !f1 f0 f a0 a. h4/ring/ring__R0 (h4/ring/ring0 a a0 f f0 f1) = a
% Assm: h4/ring/ring__case__cong: !f_27 f M_27 M. M = M_27 /\ (!a0 a1 a2 a3 a4. M_27 = h4/ring/ring0 a0 a1 a2 a3 a4 ==> f a0 a1 a2 a3 a4 = f_27 a0 a1 a2 a3 a4) ==> h4/ring/ring__CASE M f = h4/ring/ring__CASE M_27 f_27
% Assm: h4/ring/ring__accfupds_c23: !r f. h4/ring/ring__RM (h4/ring/ring__RM__fupd f r) = f (h4/ring/ring__RM r)
% Assm: h4/ring/ring__accfupds_c21: !r f. h4/ring/ring__R1 (h4/ring/ring__R1__fupd f r) = f (h4/ring/ring__R1 r)
% Assm: h4/ring/ring__accfupds_c0: !r f. h4/ring/ring__R0 (h4/ring/ring__R1__fupd f r) = h4/ring/ring__R0 r
% Assm: h4/ring/ring__accfupds_c5: !r f. h4/ring/ring__R1 (h4/ring/ring__RP__fupd f r) = h4/ring/ring__R1 r
% Assm: h4/ring/ring__fupdcanon_c2: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RP__fupd f r)
% Assm: h4/ring/EXISTS__ring: !P. (?r. P r) <=> (?a0 a f1 f0 f. P (h4/ring/ring__R0__fupd (h4/combin/K a0) (h4/ring/ring__R1__fupd (h4/combin/K a) (h4/ring/ring__RP__fupd (h4/combin/K f1) (h4/ring/ring__RM__fupd (h4/combin/K f0) (h4/ring/ring__RN__fupd (h4/combin/K f) h4/bool/ARB))))))
% Assm: h4/ring/ring__fupdfupds_c0: !r g f. h4/ring/ring__R0__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__accfupds_c9: !r f. h4/ring/ring__RP (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RP r
% Assm: h4/ring/ring__fupdfupds_c2: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__RP__fupd g r) = h4/ring/ring__RP__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__fupdcanon_c4: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RM__fupd f r)
% Assm: h4/ring/ring__fupdcanon_c0: !r g f. h4/ring/ring__R1__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__R1__fupd f r)
% Assm: h4/ring/ring__fupdcanon_c1: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__RP__fupd f r)
% Assm: h4/ring/ring__fupdcanon_c8: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RP__fupd g r) = h4/ring/ring__RP__fupd g (h4/ring/ring__RN__fupd f r)
% Assm: h4/ring/ring__nchotomy: !rr. ?a a0 f f0 f1. rr = h4/ring/ring0 a a0 f f0 f1
% Assm: h4/ring/ring__updates__eq__literal: !r f1 f0 f a0 a. h4/ring/ring__R0__fupd (h4/combin/K a0) (h4/ring/ring__R1__fupd (h4/combin/K a) (h4/ring/ring__RP__fupd (h4/combin/K f1) (h4/ring/ring__RM__fupd (h4/combin/K f0) (h4/ring/ring__RN__fupd (h4/combin/K f) r)))) = h4/ring/ring__R0__fupd (h4/combin/K a0) (h4/ring/ring__R1__fupd (h4/combin/K a) (h4/ring/ring__RP__fupd (h4/combin/K f1) (h4/ring/ring__RM__fupd (h4/combin/K f0) (h4/ring/ring__RN__fupd (h4/combin/K f) h4/bool/ARB))))
% Assm: h4/ring/ring__accfupds_c22: !r f. h4/ring/ring__RP (h4/ring/ring__RP__fupd f r) = f (h4/ring/ring__RP r)
% Assm: h4/ring/ring__accfupds_c8: !r f. h4/ring/ring__RP (h4/ring/ring__R0__fupd f r) = h4/ring/ring__RP r
% Assm: h4/ring/ring__fupdcanon_c3: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__RM__fupd f r)
% Assm: h4/ring/ring__fupdfupds_c1: !r g f. h4/ring/ring__R1__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__accfupds_c7: !r f. h4/ring/ring__R1 (h4/ring/ring__RN__fupd f r) = h4/ring/ring__R1 r
% Assm: h4/ring/ring__fupdfupds_c3: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__RM__fupd g r) = h4/ring/ring__RM__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__accfupds_c20: !r f. h4/ring/ring__R0 (h4/ring/ring__R0__fupd f r) = f (h4/ring/ring__R0 r)
% Assm: h4/ring/ring__accfupds_c1: !r f. h4/ring/ring__R0 (h4/ring/ring__RP__fupd f r) = h4/ring/ring__R0 r
% Assm: h4/ring/ring__accfupds_c11: !r f. h4/ring/ring__RP (h4/ring/ring__RN__fupd f r) = h4/ring/ring__RP r
% Assm: h4/ring/ring__case__def: !f a4 a3 a2 a1 a0. h4/ring/ring__CASE (h4/ring/ring0 a0 a1 a2 a3 a4) f = f a0 a1 a2 a3 a4
% Assm: h4/ring/ring__accfupds_c17: !r f. h4/ring/ring__RN (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RN r
% Assm: h4/ring/ring__accfupds_c4: !r f. h4/ring/ring__R1 (h4/ring/ring__R0__fupd f r) = h4/ring/ring__R1 r
% Assm: h4/ring/ring__accfupds_c18: !r f. h4/ring/ring__RN (h4/ring/ring__RP__fupd f r) = h4/ring/ring__RN r
% Assm: h4/ring/ring__accfupds_c2: !r f. h4/ring/ring__R0 (h4/ring/ring__RM__fupd f r) = h4/ring/ring__R0 r
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% 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/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Goal: !r. h4/ring/is__ring r ==> h4/semi__ring/is__semi__ring (h4/ring/semi__ring__of 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_rings_isu_u_ringu_u_def]: !r. h4/ring/is__ring r <=> (!n m. happ (happ (h4/ring/ring__RP r) n) m = happ (happ (h4/ring/ring__RP r) m) n) /\ (!n m p. happ (happ (h4/ring/ring__RP r) n) (happ (happ (h4/ring/ring__RP r) m) p) = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ring/ring__RP r) n) m)) p) /\ (!n m. happ (happ (h4/ring/ring__RM r) n) m = happ (happ (h4/ring/ring__RM r) m) n) /\ (!n m p. happ (happ (h4/ring/ring__RM r) n) (happ (happ (h4/ring/ring__RM r) m) p) = happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ring/ring__RM r) n) m)) p) /\ (!n. happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R0 r)) n = n) /\ (!n. happ (happ (h4/ring/ring__RM r) (h4/ring/ring__R1 r)) n = n) /\ (!n. happ (happ (h4/ring/ring__RP r) n) (happ (h4/ring/ring__RN r) n) = h4/ring/ring__R0 r) /\ (!n m p. happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ring/ring__RP r) n) m)) p = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ring/ring__RM r) n) p)) (happ (happ (h4/ring/ring__RM r) m) p))
% Assm [h4s_semiu_u_rings_isu_u_semiu_u_ringu_u_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 [h4s_rings_semiu_u_ringu_u_ofu_u_def]: !r. h4/ring/semi__ring__of r = h4/semi__ring/semi__ring0 (h4/ring/ring__R0 r) (h4/ring/ring__R1 r) (h4/ring/ring__RP r) (h4/ring/ring__RM r)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_TRANS]: !z y x. x = y /\ y = z ==> x = z
% Assm [h4s_rings_plusu_u_assoc]: !r. h4/ring/is__ring r ==> (!n m p. happ (happ (h4/ring/ring__RP r) n) (happ (happ (h4/ring/ring__RP r) m) p) = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ring/ring__RP r) n) m)) p)
% Assm [h4s_rings_multu_u_zerou_u_left]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RM r) (h4/ring/ring__R0 r)) n = h4/ring/ring__R0 r)
% Assm [h4s_rings_plusu_u_sym]: !r. h4/ring/is__ring r ==> (!n m. happ (happ (h4/ring/ring__RP r) n) m = happ (happ (h4/ring/ring__RP r) m) n)
% Assm [h4s_rings_multu_u_assoc]: !r. h4/ring/is__ring r ==> (!n m p. happ (happ (h4/ring/ring__RM r) n) (happ (happ (h4/ring/ring__RM r) m) p) = happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ring/ring__RM r) n) m)) p)
% Assm [h4s_rings_plusu_u_zerou_u_left]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R0 r)) n = n)
% Assm [h4s_rings_multu_u_sym]: !r. h4/ring/is__ring r ==> (!n m. happ (happ (h4/ring/ring__RM r) n) m = happ (happ (h4/ring/ring__RM r) m) n)
% Assm [h4s_rings_oppu_u_def]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RP r) n) (happ (h4/ring/ring__RN r) n) = h4/ring/ring__R0 r)
% Assm [h4s_rings_multu_u_oneu_u_left]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RM r) (h4/ring/ring__R1 r)) n = n)
% Assm [h4s_rings_plusu_u_zerou_u_right]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RP r) n) (h4/ring/ring__R0 r) = n)
% Assm [h4s_rings_multu_u_zerou_u_right]: !r. h4/ring/is__ring r ==> (!n. happ (happ (h4/ring/ring__RM r) n) (h4/ring/ring__R0 r) = h4/ring/ring__R0 r)
% Assm [h4s_rings_distru_u_left]: !r. h4/ring/is__ring r ==> (!n m p. happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ring/ring__RP r) n) m)) p = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ring/ring__RM r) n) p)) (happ (happ (h4/ring/ring__RM r) m) p))
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_rings_ringu_u_repfnsu_c0]: !a. h4/ring/_20_40ind__typering1 (h4/ring/_20_40ind__typering2 a) = a
% Assm [h4s_rings_ring1]: h4/ring/ring0 = h4/ring/_20_40ind__typering0
% Assm [h4s_rings_ringu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 (h4/pair/_2C a3 a4)))) _0) ==> happ _27ring_27 a0_270) ==> happ _27ring_27 r) <=> h4/ring/_20_40ind__typering2 (h4/ring/_20_40ind__typering1 r) = r)
% Assm [h4s_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typering0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x' x x' x. happ (happ (happ (happ (happ h4/ring/_20_40ind__typering0 x) x') x) x') x = h4/ring/_20_40ind__typering1 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x (h4/pair/_2C x' (h4/pair/_2C x (h4/pair/_2C x' x)))) _0))
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_semiu_u_rings_plusu_u_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 [h4s_semiu_u_rings_distru_u_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 [h4s_semiu_u_rings_multu_u_oneu_u_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 [h4s_semiu_u_rings_plusu_u_zerou_u_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 [h4s_semiu_u_rings_plusu_u_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 [h4s_semiu_u_rings_multu_u_rotate]: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r m n) p = h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r n p) m)
% Assm [h4s_semiu_u_rings_multu_u_permute]: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r m n) p = h4/semi__ring/semi__ring__SRM r (h4/semi__ring/semi__ring__SRM r m p) n)
% Assm [h4s_semiu_u_rings_multu_u_zerou_u_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 [h4s_semiu_u_rings_multu_u_zerou_u_right]: !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)
% Assm [h4s_semiu_u_rings_multu_u_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 [h4s_semiu_u_rings_multu_u_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 [h4s_semiu_u_rings_multu_u_oneu_u_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 [h4s_semiu_u_rings_plusu_u_zerou_u_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 [h4s_semiu_u_rings_plusu_u_permute]: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r m n) p = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r m p) n)
% Assm [h4s_semiu_u_rings_distru_u_right]: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRM r m (h4/semi__ring/semi__ring__SRP r n p) = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRM r m n) (h4/semi__ring/semi__ring__SRM r m p))
% Assm [h4s_semiu_u_rings_plusu_u_rotate]: !r. h4/semi__ring/is__semi__ring r ==> (!m n p. h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r m n) p = h4/semi__ring/semi__ring__SRP r (h4/semi__ring/semi__ring__SRP r n p) m)
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c1]: !f2 f1 f0 f a0 a. h4/ring/ring__R1__fupd f2 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) (happ f2 a0)) f) f0) f1
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c3]: !f2 f1 f0 f a0 a. h4/ring/ring__RM__fupd f2 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) (happ f2 f0)) f1
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c0]: !f2 f1 f0 f a0 a. h4/ring/ring__R0__fupd f2 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 (happ f2 a)) a0) f) f0) f1
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c4]: !f2 f1 f0 f a0 a. h4/ring/ring__RN__fupd f2 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) (happ f2 f1)
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c2]: !f2 f1 f0 f a0 a. h4/ring/ring__RP__fupd f2 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) (happ f2 f)) f0) f1
% Assm [h4s_rings_ringu_u_accessorsu_c2]: !f1 f0 f a0 a. h4/ring/ring__RP (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_rings_ringu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 (h4/pair/_2C a3 a4)))) _1) ==> happ _27ring_27 a0_270) ==> happ _27ring_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_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_rings_ringu_u_accessorsu_c4]: !f1 f0 f a0 a. h4/ring/ring__RN (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f1
% Assm [h4s_rings_ringu_u_accessorsu_c3]: !f1 f0 f a0 a. h4/ring/ring__RM (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f0
% Assm [h4s_rings_ringu_u_accfupdsu_c19]: !r f. h4/ring/ring__RN (h4/ring/ring__RM__fupd f r) = h4/ring/ring__RN r
% Assm [h4s_rings_ringu_u_accfupdsu_c10]: !r f. h4/ring/ring__RP (h4/ring/ring__RM__fupd f r) = h4/ring/ring__RP r
% Assm [h4s_rings_ringu_u_accfupdsu_c13]: !r f. h4/ring/ring__RM (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RM r
% Assm [h4s_rings_ringu_u_induction]: !P. (!a a0 f f0 f1. happ P (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1)) ==> (!r. happ P r)
% Assm [h4s_rings_ringu_u_accfupdsu_c12]: !r f. h4/ring/ring__RM (h4/ring/ring__R0__fupd f r) = h4/ring/ring__RM r
% Assm [h4s_rings_ringu_u_fupdfupdsu_c4]: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RN__fupd g r) = h4/ring/ring__RN__fupd (h4/combin/o f g) r
% Assm [h4s_rings_ringu_u_fupdcanonu_c7]: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RN__fupd f r)
% Assm [h4s_rings_ringu_u_fupdcanonu_c9]: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RM__fupd g r) = h4/ring/ring__RM__fupd g (h4/ring/ring__RN__fupd f r)
% Assm [h4s_rings_ringu_u_accfupdsu_c15]: !r f. h4/ring/ring__RM (h4/ring/ring__RN__fupd f r) = h4/ring/ring__RM r
% Assm [h4s_rings_ringu_u_accfupdsu_c24]: !r f. h4/ring/ring__RN (h4/ring/ring__RN__fupd f r) = happ f (h4/ring/ring__RN r)
% Assm [h4s_rings_ringu_u_accessorsu_c1]: !f1 f0 f a0 a. h4/ring/ring__R1 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = a0
% Assm [h4s_rings_ringu_u_accessorsu_c0]: !f1 f0 f a0 a. h4/ring/ring__R0 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = a
% Assm [h4s_rings_ringu_u_caseu_u_cong]: !f_27 f M_27 M. M = M_27 /\ (!a0 a1 a2 a3 a4. M_27 = happ (happ (happ (happ (happ h4/ring/ring0 a0) a1) a2) a3) a4 ==> happ (happ (happ (happ (happ f a0) a1) a2) a3) a4 = happ (happ (happ (happ (happ f_27 a0) a1) a2) a3) a4) ==> h4/ring/ring__CASE M f = h4/ring/ring__CASE M_27 f_27
% Assm [h4s_rings_ringu_u_accfupdsu_c23]: !r f. h4/ring/ring__RM (h4/ring/ring__RM__fupd f r) = happ f (h4/ring/ring__RM r)
% Assm [h4s_rings_ringu_u_accfupdsu_c21]: !r f. h4/ring/ring__R1 (h4/ring/ring__R1__fupd f r) = happ f (h4/ring/ring__R1 r)
% Assm [h4s_rings_ringu_u_accfupdsu_c0]: !r f. h4/ring/ring__R0 (h4/ring/ring__R1__fupd f r) = h4/ring/ring__R0 r
% Assm [h4s_rings_ringu_u_accfupdsu_c5]: !r f. h4/ring/ring__R1 (h4/ring/ring__RP__fupd f r) = h4/ring/ring__R1 r
% Assm [h4s_rings_ringu_u_fupdcanonu_c2]: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RP__fupd f r)
% Assm [h4s_rings_EXISTSu_u_ring]: !P. (?r. happ P r) <=> (?a0 a f1 f0 f. happ P (h4/ring/ring__R0__fupd (h4/combin/K a0) (h4/ring/ring__R1__fupd (h4/combin/K a) (h4/ring/ring__RP__fupd (h4/combin/K f1) (h4/ring/ring__RM__fupd (h4/combin/K f0) (h4/ring/ring__RN__fupd (h4/combin/K f) h4/bool/ARB))))))
% Assm [h4s_rings_ringu_u_fupdfupdsu_c0]: !r g f. h4/ring/ring__R0__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd (h4/combin/o f g) r
% Assm [h4s_rings_ringu_u_accfupdsu_c9]: !r f. h4/ring/ring__RP (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RP r
% Assm [h4s_rings_ringu_u_fupdfupdsu_c2]: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__RP__fupd g r) = h4/ring/ring__RP__fupd (h4/combin/o f g) r
% Assm [h4s_rings_ringu_u_fupdcanonu_c4]: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RM__fupd f r)
% Assm [h4s_rings_ringu_u_fupdcanonu_c0]: !r g f. h4/ring/ring__R1__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__R1__fupd f r)
% Assm [h4s_rings_ringu_u_fupdcanonu_c1]: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__RP__fupd f r)
% Assm [h4s_rings_ringu_u_fupdcanonu_c8]: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RP__fupd g r) = h4/ring/ring__RP__fupd g (h4/ring/ring__RN__fupd f r)
% Assm [h4s_rings_ringu_u_nchotomy]: !rr. ?a a0 f f0 f1. rr = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1
% Assm [h4s_rings_ringu_u_updatesu_u_equ_u_literal]: !r f1 f0 f a0 a. h4/ring/ring__R0__fupd (h4/combin/K a0) (h4/ring/ring__R1__fupd (h4/combin/K a) (h4/ring/ring__RP__fupd (h4/combin/K f1) (h4/ring/ring__RM__fupd (h4/combin/K f0) (h4/ring/ring__RN__fupd (h4/combin/K f) r)))) = h4/ring/ring__R0__fupd (h4/combin/K a0) (h4/ring/ring__R1__fupd (h4/combin/K a) (h4/ring/ring__RP__fupd (h4/combin/K f1) (h4/ring/ring__RM__fupd (h4/combin/K f0) (h4/ring/ring__RN__fupd (h4/combin/K f) h4/bool/ARB))))
% Assm [h4s_rings_ringu_u_accfupdsu_c22]: !r f. h4/ring/ring__RP (h4/ring/ring__RP__fupd f r) = happ f (h4/ring/ring__RP r)
% Assm [h4s_rings_ringu_u_accfupdsu_c8]: !r f. h4/ring/ring__RP (h4/ring/ring__R0__fupd f r) = h4/ring/ring__RP r
% Assm [h4s_rings_ringu_u_fupdcanonu_c3]: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__RM__fupd f r)
% Assm [h4s_rings_ringu_u_fupdfupdsu_c1]: !r g f. h4/ring/ring__R1__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd (h4/combin/o f g) r
% Assm [h4s_rings_ringu_u_accfupdsu_c7]: !r f. h4/ring/ring__R1 (h4/ring/ring__RN__fupd f r) = h4/ring/ring__R1 r
% Assm [h4s_rings_ringu_u_fupdfupdsu_c3]: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__RM__fupd g r) = h4/ring/ring__RM__fupd (h4/combin/o f g) r
% Assm [h4s_rings_ringu_u_accfupdsu_c20]: !r f. h4/ring/ring__R0 (h4/ring/ring__R0__fupd f r) = happ f (h4/ring/ring__R0 r)
% Assm [h4s_rings_ringu_u_accfupdsu_c1]: !r f. h4/ring/ring__R0 (h4/ring/ring__RP__fupd f r) = h4/ring/ring__R0 r
% Assm [h4s_rings_ringu_u_accfupdsu_c11]: !r f. h4/ring/ring__RP (h4/ring/ring__RN__fupd f r) = h4/ring/ring__RP r
% Assm [h4s_rings_ringu_u_caseu_u_def]: !f a4 a3 a2 a1 a0. h4/ring/ring__CASE (happ (happ (happ (happ (happ h4/ring/ring0 a0) a1) a2) a3) a4) f = happ (happ (happ (happ (happ f a0) a1) a2) a3) a4
% Assm [h4s_rings_ringu_u_accfupdsu_c17]: !r f. h4/ring/ring__RN (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RN r
% Assm [h4s_rings_ringu_u_accfupdsu_c4]: !r f. h4/ring/ring__R1 (h4/ring/ring__R0__fupd f r) = h4/ring/ring__R1 r
% Assm [h4s_rings_ringu_u_accfupdsu_c18]: !r f. h4/ring/ring__RN (h4/ring/ring__RP__fupd f r) = h4/ring/ring__RN r
% Assm [h4s_rings_ringu_u_accfupdsu_c2]: !r f. h4/ring/ring__R0 (h4/ring/ring__RM__fupd f r) = h4/ring/ring__R0 r
% Assm [h4s_combins_Ku_u_THM]: !y x. happ (h4/combin/K x) y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% 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_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Goal: !r. h4/ring/is__ring r ==> h4/semi__ring/is__semi__ring (h4/ring/semi__ring__of 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_Q1391338,TV_Q1391334]: ![V_f, V_g]: (![V_x]: s(TV_Q1391334,happ(s(t_fun(TV_Q1391338,TV_Q1391334),V_f),s(TV_Q1391338,V_x))) = s(TV_Q1391334,happ(s(t_fun(TV_Q1391338,TV_Q1391334),V_g),s(TV_Q1391338,V_x))) => s(t_fun(TV_Q1391338,TV_Q1391334),V_f) = s(t_fun(TV_Q1391338,TV_Q1391334),V_g))).
fof(ah4s_rings_isu_u_ringu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) <=> (![V_n, V_m]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_n))) & (![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) & (![V_n, V_m]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_n))) & (![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) & (![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n) & (![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n) & (![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))) & ![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_p))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p)))))))))))))).
fof(ah4s_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,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,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,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_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,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_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,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,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,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_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,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_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,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,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,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,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_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,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_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,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_rings_semiu_u_ringu_u_ofu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_TRANS, axiom, ![TV_u_27a]: ![V_z, V_y, V_x]: ((s(TV_u_27a,V_x) = s(TV_u_27a,V_y) & s(TV_u_27a,V_y) = s(TV_u_27a,V_z)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_z))).
fof(ah4s_rings_plusu_u_assoc, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))))).
fof(ah4s_rings_multu_u_zerou_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))).
fof(ah4s_rings_plusu_u_sym, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n, V_m]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_n))))).
fof(ah4s_rings_multu_u_assoc, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))))).
fof(ah4s_rings_plusu_u_zerou_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n))).
fof(ah4s_rings_multu_u_sym, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n, V_m]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_n))))).
fof(ah4s_rings_oppu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))).
fof(ah4s_rings_multu_u_oneu_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n))).
fof(ah4s_rings_plusu_u_zerou_u_right, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,V_n))).
fof(ah4s_rings_multu_u_zerou_u_right, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))).
fof(ah4s_rings_distru_u_left, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n, V_m, V_p]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_m))))),s(TV_u_27a,V_p))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))),s(TV_u_27a,V_p))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))))))).
fof(ah4s_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_rings_ringu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering2(s(t_h4s_rings_ring(TV_u_27a),V_a))))) = s(t_h4s_rings_ring(TV_u_27a),V_a)).
fof(ah4s_rings_ring1, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering0)).
fof(ah4s_rings_ringu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3, V_a4]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3),s(t_fun(TV_u_27a,TV_u_27a),V_a4))))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering2(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r)))).
fof(ah4s_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typering0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_x, V_xi_, V_x0, V_xi_0, V_x1]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering0),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0))),s(t_fun(TV_u_27a,TV_u_27a),V_x1))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_xi_),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0),s(t_fun(TV_u_27a,TV_u_27a),V_x1))))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))))))).
fof(ah4s_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_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,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_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,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_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_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,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_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,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_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,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,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_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,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_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,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,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_multu_u_rotate, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_m, V_n, V_p]: s(TV_u_27a,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_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_n))),s(TV_u_27a,V_p))) = s(TV_u_27a,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_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,V_p))),s(TV_u_27a,V_m))))).
fof(ah4s_semiu_u_rings_multu_u_permute, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_m, V_n, V_p]: s(TV_u_27a,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_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_n))),s(TV_u_27a,V_p))) = s(TV_u_27a,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_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,V_n))))).
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,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_multu_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,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))))).
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,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,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,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_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,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_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_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,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_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,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_permute, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_m, V_n, V_p]: s(TV_u_27a,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_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_n))),s(TV_u_27a,V_p))) = s(TV_u_27a,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_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_p))),s(TV_u_27a,V_n))))).
fof(ah4s_semiu_u_rings_distru_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_m, V_n, V_p]: s(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,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_p))))) = s(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_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_n))),s(TV_u_27a,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_plusu_u_rotate, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_m, V_n, V_p]: s(TV_u_27a,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_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_n))),s(TV_u_27a,V_p))) = s(TV_u_27a,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_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_p))),s(TV_u_27a,V_m))))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c1, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f2),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f2),s(TV_u_27a,V_a0))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c3, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_f2),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),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_f2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c0, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f2),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f2),s(TV_u_27a,V_a))))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c4, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f2),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f2),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c2, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_f2),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),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_f2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
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_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_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_rings_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_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_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_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_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_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_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_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))) <=> ![V_uu_27ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3, V_a4]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3),s(t_fun(TV_u_27a,TV_u_27a),V_a4))))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_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_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_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_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),s(t_fun(t_h4s_rings_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_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_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_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_rings_ringu_u_accessorsu_c4, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,TV_u_27a),V_f1)).
fof(ah4s_rings_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_rings_ringu_u_accfupdsu_c19, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c10, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c13, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_induction, axiom, ![TV_u_27a]: ![V_P]: (![V_a, V_a0, V_f, V_f0, V_f1]: p(s(t_bool,happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_bool),V_P),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))))) => ![V_r]: p(s(t_bool,happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_bool),V_P),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c12, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c4, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c7, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_c9, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c15, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c24, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a0)).
fof(ah4s_rings_ringu_u_accessorsu_c0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a)).
fof(ah4s_rings_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_rings_ring(TV_u_27a),V_M) = s(t_h4s_rings_ring(TV_u_27a),V_Mu_27) & ![V_a0, V_a1, V_a2, V_a3, V_a4]: (s(t_h4s_rings_ring(TV_u_27a),V_Mu_27) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_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,TV_u_27a),V_a4))) => s(TV_u_27b,happ(s(t_fun(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,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)),t_fun(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)),t_fun(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)),t_fun(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(t_fun(TV_u_27a,TV_u_27a),V_a4))) = s(TV_u_27b,happ(s(t_fun(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,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)),t_fun(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)),t_fun(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)),t_fun(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(t_fun(TV_u_27a,TV_u_27a),V_a4))))) => s(TV_u_27b,h4s_rings_ringu_u_case(s(t_h4s_rings_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)),t_fun(t_fun(TV_u_27a,TV_u_27a),TV_u_27b))))),V_f))) = s(TV_u_27b,h4s_rings_ringu_u_case(s(t_h4s_rings_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)),t_fun(t_fun(TV_u_27a,TV_u_27a),TV_u_27b))))),V_fu_27))))).
fof(ah4s_rings_ringu_u_accfupdsu_c23, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r))))) = 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_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c21, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c0, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c5, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c2, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_EXISTSu_u_ring, axiom, ![TV_u_27a]: ![V_P]: (?[V_r]: p(s(t_bool,happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_bool),V_P),s(t_h4s_rings_ring(TV_u_27a),V_r)))) <=> ?[V_a0, V_a, V_f1, V_f0, V_f]: p(s(t_bool,happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_bool),V_P),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a0))),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a))),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),h4s_combins_k(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),h4s_bools_arb)))))))))))))))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c0, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_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_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c9, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c2, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c4, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_c0, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_c1, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_c8, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_nchotomy, axiom, ![TV_u_27a]: ![V_rr]: ?[V_a, V_a0, V_f, V_f0, V_f1]: s(t_h4s_rings_ring(TV_u_27a),V_rr) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_updatesu_u_equ_u_literal, axiom, ![TV_u_27a]: ![V_r, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a0))),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a))),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),h4s_combins_k(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))))))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a0))),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_a))),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),h4s_combins_k(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),h4s_bools_arb)))))))))))).
fof(ah4s_rings_ringu_u_accfupdsu_c22, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r))))) = 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_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c8, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c3, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c1, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_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_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c7, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c3, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c20, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c1, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c11, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_caseu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_a4, V_a3, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_rings_ringu_u_case(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_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,TV_u_27a),V_a4))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),TV_u_27b))))),V_f))) = s(TV_u_27b,happ(s(t_fun(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,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)),t_fun(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)),t_fun(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)),t_fun(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(t_fun(TV_u_27a,TV_u_27a),V_a4)))).
fof(ah4s_rings_ringu_u_accfupdsu_c17, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c4, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c18, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rpu_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_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c2, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ringu_u_rmu_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_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_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_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_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_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(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_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_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(ch4s_rings_ringu_u_isu_u_semiu_u_ring, conjecture, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))))).
