%   ORIGINAL: h4/semi__ring/semi__ring__fupdcanon_c4
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/semi__ring/semi__ring__repfns_c1: !r. (\a0_27. !_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3) ==> _27semi__ring_27 a0_270) ==> _27semi__ring_27 a0_27) r <=> h4/semi__ring/_20_40ind__typesemi__ring2 (h4/semi__ring/_20_40ind__typesemi__ring1 r) = r
% Assm: h4/semi__ring/hidden____20__40ind____typesemi____ring0____def: h4/semi__ring/_20_40ind__typesemi__ring0 = (\a0 a1 a2 a3. h4/semi__ring/_20_40ind__typesemi__ring1 ((\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3))
% Assm: h4/semi__ring/semi__ring1: h4/semi__ring/semi__ring0 = h4/semi__ring/_20_40ind__typesemi__ring0
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/semi__ring/semi__ring__repfns_c0: !a. h4/semi__ring/_20_40ind__typesemi__ring1 (h4/semi__ring/_20_40ind__typesemi__ring2 a) = a
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/semi__ring/semi__ring__fn__updates_c3: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SRM__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a a0 f (f1 f0)
% Assm: h4/semi__ring/semi__ring__fn__updates_c1: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SR1__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a (f1 a0) f f0
% Assm: h4/semi__ring/semi__ring__SRM__fupd0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SRM__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a a0 f (f1 f0)
% Assm: h4/semi__ring/semi__ring__SR1__fupd0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SR1__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a (f1 a0) f f0
% Assm: h4/semi__ring/semi__ring__fupdcanon_c3: !s g f. h4/semi__ring/semi__ring__SRM__fupd f (h4/semi__ring/semi__ring__SR0__fupd g s) = h4/semi__ring/semi__ring__SR0__fupd g (h4/semi__ring/semi__ring__SRM__fupd f s)
% Assm: h4/semi__ring/semi__ring__accfupds_c8: !s f. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring__SRM__fupd f s) = h4/semi__ring/semi__ring__SRP s
% Assm: h4/semi__ring/semi__ring__fupdfupds_c3: !s g f. h4/semi__ring/semi__ring__SRM__fupd f (h4/semi__ring/semi__ring__SRM__fupd g s) = h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g) s
% Assm: h4/semi__ring/semi__ring__accfupds_c5: !s f. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring__SRM__fupd f s) = h4/semi__ring/semi__ring__SR1 s
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/semi__ring/semi__ring__accfupds_c2: !s f. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring__SRM__fupd f s) = h4/semi__ring/semi__ring__SR0 s
% Assm: h4/semi__ring/semi__ring__accfupds_c15: !s f. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring__SRM__fupd f s) = f (h4/semi__ring/semi__ring__SRM s)
% Assm: h4/semi__ring/semi__ring__fupdcanon_c2: !s g f. h4/semi__ring/semi__ring__SRP__fupd f (h4/semi__ring/semi__ring__SR1__fupd g s) = h4/semi__ring/semi__ring__SR1__fupd g (h4/semi__ring/semi__ring__SRP__fupd f s)
% Assm: h4/semi__ring/semi__ring__fn__updates_c2: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SRP__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a a0 (f1 f) f0
% Assm: h4/semi__ring/semi__ring__fn__updates_c0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SR0__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 (f1 a) a0 f f0
% Assm: h4/semi__ring/semi__ring__accfupds_c7: !s f. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring__SR1__fupd f s) = h4/semi__ring/semi__ring__SRP s
% Assm: h4/semi__ring/semi__ring__accfupds_c13: !s f. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring__SR1__fupd f s) = f (h4/semi__ring/semi__ring__SR1 s)
% Assm: h4/semi__ring/semi__ring__accfupds_c10: !s f. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring__SR1__fupd f s) = h4/semi__ring/semi__ring__SRM s
% Assm: h4/semi__ring/semi__ring__fupdfupds_c1: !s g f. h4/semi__ring/semi__ring__SR1__fupd f (h4/semi__ring/semi__ring__SR1__fupd g s) = h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g) s
% Assm: h4/semi__ring/semi__ring__fupdcanon_c0: !s g f. h4/semi__ring/semi__ring__SR1__fupd f (h4/semi__ring/semi__ring__SR0__fupd g s) = h4/semi__ring/semi__ring__SR0__fupd g (h4/semi__ring/semi__ring__SR1__fupd f s)
% Assm: h4/semi__ring/semi__ring__accfupds_c0: !s f. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring__SR1__fupd f s) = h4/semi__ring/semi__ring__SR0 s
% Assm: h4/semi__ring/semi__ring__accessors_c2: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm: h4/semi__ring/semi__ring__accessors_c1: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring0 a a0 f f0) = a0
% Assm: h4/semi__ring/semi__ring__accessors_c0: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/semi__ring/semi__ring__accessors_c3: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm: h4/semi__ring/semi__ring__SRP__fupd0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SRP__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 a a0 (f1 f) f0
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c6: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SRM__fupd g) = h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c2: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c7: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)) h
% Assm: h4/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/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% 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/semi__ring/semi__ring__fupdcanon_c1: !s g f. h4/semi__ring/semi__ring__SRP__fupd f (h4/semi__ring/semi__ring__SR0__fupd g s) = h4/semi__ring/semi__ring__SR0__fupd g (h4/semi__ring/semi__ring__SRP__fupd f s)
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c3: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)) h
% Assm: h4/semi__ring/semi__ring__accfupds_c1: !s f. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring__SRP__fupd f s) = h4/semi__ring/semi__ring__SR0 s
% Assm: h4/semi__ring/semi__ring__accfupds_c14: !s f. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring__SRP__fupd f s) = f (h4/semi__ring/semi__ring__SRP s)
% Assm: h4/semi__ring/semi__ring__fupdfupds_c2: !s g f. h4/semi__ring/semi__ring__SRP__fupd f (h4/semi__ring/semi__ring__SRP__fupd g s) = h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g) s
% Assm: h4/semi__ring/semi__ring__accfupds_c11: !s f. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring__SRP__fupd f s) = h4/semi__ring/semi__ring__SRM s
% Assm: h4/semi__ring/semi__ring__accfupds_c4: !s f. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring__SRP__fupd f s) = h4/semi__ring/semi__ring__SR1 s
% Assm: h4/semi__ring/semi__ring__SR0__fupd0: !f1 f0 f a0 a. h4/semi__ring/semi__ring__SR0__fupd f1 (h4/semi__ring/semi__ring0 a a0 f f0) = h4/semi__ring/semi__ring0 (f1 a) a0 f f0
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c4: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/semi__ring/semi__ring__SRP__fupd g) = h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g)
% Assm: h4/semi__ring/semi__ring__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3) ==> _27semi__ring_27 a0_270) ==> _27semi__ring_27 a0_27) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/semi__ring/semi__ring__accfupds_c3: !s f. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring__SR0__fupd f s) = h4/semi__ring/semi__ring__SR1 s
% Assm: h4/semi__ring/semi__ring__accfupds_c6: !s f. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring__SR0__fupd f s) = h4/semi__ring/semi__ring__SRP s
% Assm: h4/semi__ring/semi__ring__accfupds_c9: !s f. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring__SR0__fupd f s) = h4/semi__ring/semi__ring__SRM s
% Assm: h4/semi__ring/semi__ring__accfupds_c12: !s f. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring__SR0__fupd f s) = f (h4/semi__ring/semi__ring__SR0 s)
% Assm: h4/semi__ring/semi__ring__fupdfupds_c0: !s g f. h4/semi__ring/semi__ring__SR0__fupd f (h4/semi__ring/semi__ring__SR0__fupd g s) = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g) s
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/combin/o__ASSOC: !h g f. h4/combin/o f (h4/combin/o g h) = h4/combin/o (h4/combin/o f g) h
% Assm: h4/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/semi__ring/semi__ring__fupdfupds__comp_c0: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g)
% Assm: h4/arithmetic/FUNPOW0_c1: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (f x)
% Assm: h4/arithmetic/FUNPOW0_c0: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm: h4/arithmetic/FUNPOW__0: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm: h4/prim__rec/SIMP__REC__REL__UNIQUE: !x m2 m1 g2 g1 f. h4/prim__rec/SIMP__REC__REL g1 x f m1 /\ h4/prim__rec/SIMP__REC__REL g2 x f m2 ==> (!n. h4/prim__rec/_3C n m1 /\ h4/prim__rec/_3C n m2 ==> g1 n = g2 n)
% Assm: h4/while/OWHILE__WHILE: !s_27 s f G. h4/while/OWHILE G f s = h4/option/SOME s_27 ==> h4/while/WHILE G f s = s_27
% Assm: h4/relation/IDEM0: !f. h4/relation/IDEM f <=> (!x. f (f x) = f x)
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/relation/INVOL0: !f. h4/relation/INVOL f <=> (!x. f (f x) = x)
% Assm: h4/arithmetic/FUNPOW__SUC: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = f (h4/arithmetic/FUNPOW f n x)
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/while/OWHILE__ENDCOND: !s_27 s f G. h4/while/OWHILE G f s = h4/option/SOME s_27 ==> ~G s_27
% Assm: h4/relation/INVOL__ONE__ENO: !f. h4/relation/INVOL f ==> (!a b. f a = b <=> a = f b)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/relation/IDEM__DEF: !f. h4/relation/IDEM f <=> h4/combin/o f f = f
% Assm: h4/prim__rec/SIMP__REC__REL0: !x n fun f. h4/prim__rec/SIMP__REC__REL fun x f n <=> fun h4/num/0 = x /\ (!m. h4/prim__rec/_3C m n ==> fun (h4/num/SUC m) = f (fun m))
% Assm: h4/prim__rec/SUC__LESS: !n m. h4/prim__rec/_3C (h4/num/SUC m) n ==> h4/prim__rec/_3C m n
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/relation/INVOL__DEF: !f. h4/relation/INVOL f <=> h4/combin/o f f = h4/combin/I
% Assm: h4/while/OWHILE__def: !s f G. h4/while/OWHILE G f s = h4/bool/COND (?n. ~G (h4/arithmetic/FUNPOW f n s)) (h4/option/SOME (h4/arithmetic/FUNPOW f (h4/while/LEAST (\n. ~G (h4/arithmetic/FUNPOW f n s))) s)) h4/option/NONE
% Assm: h4/option/IF__EQUALS__OPTION_c2: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/while/LEAST__ELIM: !Q P. (?n. P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~P m) /\ P n ==> Q n) ==> Q (h4/while/LEAST P)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/while/WHILE0: !x g P. h4/while/WHILE P g x = h4/bool/COND (P x) (h4/while/WHILE P g (g x)) x
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% 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/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Goal: !s g f. h4/semi__ring/semi__ring__SRM__fupd f (h4/semi__ring/semi__ring__SR1__fupd g s) = h4/semi__ring/semi__ring__SR1__fupd g (h4/semi__ring/semi__ring__SRM__fupd f s)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 a3))) _0) ==> happ _27semi__ring_27 a0_270) ==> happ _27semi__ring_27 r) <=> h4/semi__ring/_20_40ind__typesemi__ring2 (h4/semi__ring/_20_40ind__typesemi__ring1 r) = r)
% Assm [h4s_semiu_u_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesemiu_u_u_u_ring0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x' x x'. happ (happ (happ (happ h4/semi__ring/_20_40ind__typesemi__ring0 x) x') x) x' = h4/semi__ring/_20_40ind__typesemi__ring1 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x (h4/pair/_2C x' (h4/pair/_2C x x'))) _0))
% Assm [h4s_semiu_u_rings_semiu_u_ring1]: h4/semi__ring/semi__ring0 = h4/semi__ring/_20_40ind__typesemi__ring0
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c0]: !a. h4/semi__ring/_20_40ind__typesemi__ring1 (h4/semi__ring/_20_40ind__typesemi__ring2 a) = a
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c3]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SRM__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) (happ f1 f0)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c1]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SR1__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) (happ f1 a0)) f) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SRMu_u_fupd0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SRM__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) (happ f1 f0)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SR1u_u_fupd0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SR1__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) (happ f1 a0)) f) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c3]: !s g f. happ (h4/semi__ring/semi__ring__SRM__fupd f) (happ (h4/semi__ring/semi__ring__SR0__fupd g) s) = happ (h4/semi__ring/semi__ring__SR0__fupd g) (happ (h4/semi__ring/semi__ring__SRM__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c8]: !s f. h4/semi__ring/semi__ring__SRP (happ (h4/semi__ring/semi__ring__SRM__fupd f) s) = h4/semi__ring/semi__ring__SRP s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c3]: !s g f. happ (h4/semi__ring/semi__ring__SRM__fupd f) (happ (h4/semi__ring/semi__ring__SRM__fupd g) s) = happ (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)) s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c5]: !s f. h4/semi__ring/semi__ring__SR1 (happ (h4/semi__ring/semi__ring__SRM__fupd f) s) = h4/semi__ring/semi__ring__SR1 s
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c2]: !s f. h4/semi__ring/semi__ring__SR0 (happ (h4/semi__ring/semi__ring__SRM__fupd f) s) = h4/semi__ring/semi__ring__SR0 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c15]: !s f. h4/semi__ring/semi__ring__SRM (happ (h4/semi__ring/semi__ring__SRM__fupd f) s) = happ f (h4/semi__ring/semi__ring__SRM s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c2]: !s g f. happ (h4/semi__ring/semi__ring__SRP__fupd f) (happ (h4/semi__ring/semi__ring__SR1__fupd g) s) = happ (h4/semi__ring/semi__ring__SR1__fupd g) (happ (h4/semi__ring/semi__ring__SRP__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c2]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SRP__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) (happ f1 f)) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SR0__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 (happ f1 a)) a0) f) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c7]: !s f. h4/semi__ring/semi__ring__SRP (happ (h4/semi__ring/semi__ring__SR1__fupd f) s) = h4/semi__ring/semi__ring__SRP s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c13]: !s f. h4/semi__ring/semi__ring__SR1 (happ (h4/semi__ring/semi__ring__SR1__fupd f) s) = happ f (h4/semi__ring/semi__ring__SR1 s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c10]: !s f. h4/semi__ring/semi__ring__SRM (happ (h4/semi__ring/semi__ring__SR1__fupd f) s) = h4/semi__ring/semi__ring__SRM s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c1]: !s g f. happ (h4/semi__ring/semi__ring__SR1__fupd f) (happ (h4/semi__ring/semi__ring__SR1__fupd g) s) = happ (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)) s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c0]: !s g f. happ (h4/semi__ring/semi__ring__SR1__fupd f) (happ (h4/semi__ring/semi__ring__SR0__fupd g) s) = happ (h4/semi__ring/semi__ring__SR0__fupd g) (happ (h4/semi__ring/semi__ring__SR1__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c0]: !s f. h4/semi__ring/semi__ring__SR0 (happ (h4/semi__ring/semi__ring__SR1__fupd f) s) = h4/semi__ring/semi__ring__SR0 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c1]: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SRPu_u_fupd0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SRP__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) (happ f1 f)) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c6]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SRM__fupd g) = h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c2]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c7]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd (h4/combin/o f g)) h
% Assm [h4s_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_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% 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_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c1]: !s g f. happ (h4/semi__ring/semi__ring__SRP__fupd f) (happ (h4/semi__ring/semi__ring__SR0__fupd g) s) = happ (h4/semi__ring/semi__ring__SR0__fupd g) (happ (h4/semi__ring/semi__ring__SRP__fupd f) s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c3]: !h g f. h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd f) (h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) h) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd (h4/combin/o f g)) h
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c1]: !s f. h4/semi__ring/semi__ring__SR0 (happ (h4/semi__ring/semi__ring__SRP__fupd f) s) = h4/semi__ring/semi__ring__SR0 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c14]: !s f. h4/semi__ring/semi__ring__SRP (happ (h4/semi__ring/semi__ring__SRP__fupd f) s) = happ f (h4/semi__ring/semi__ring__SRP s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c2]: !s g f. happ (h4/semi__ring/semi__ring__SRP__fupd f) (happ (h4/semi__ring/semi__ring__SRP__fupd g) s) = happ (h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g)) s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c11]: !s f. h4/semi__ring/semi__ring__SRM (happ (h4/semi__ring/semi__ring__SRP__fupd f) s) = h4/semi__ring/semi__ring__SRM s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c4]: !s f. h4/semi__ring/semi__ring__SR1 (happ (h4/semi__ring/semi__ring__SRP__fupd f) s) = h4/semi__ring/semi__ring__SR1 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SR0u_u_fupd0]: !f1 f0 f a0 a. happ (h4/semi__ring/semi__ring__SR0__fupd f1) (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = happ (happ (happ (happ h4/semi__ring/semi__ring0 (happ f1 a)) a0) f) f0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c4]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRP__fupd f) (h4/semi__ring/semi__ring__SRP__fupd g) = h4/semi__ring/semi__ring__SRP__fupd (h4/combin/o f g)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 a3))) _1) ==> happ _27semi__ring_27 a0_270) ==> happ _27semi__ring_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c3]: !s f. h4/semi__ring/semi__ring__SR1 (happ (h4/semi__ring/semi__ring__SR0__fupd f) s) = h4/semi__ring/semi__ring__SR1 s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c6]: !s f. h4/semi__ring/semi__ring__SRP (happ (h4/semi__ring/semi__ring__SR0__fupd f) s) = h4/semi__ring/semi__ring__SRP s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c9]: !s f. h4/semi__ring/semi__ring__SRM (happ (h4/semi__ring/semi__ring__SR0__fupd f) s) = h4/semi__ring/semi__ring__SRM s
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c12]: !s f. h4/semi__ring/semi__ring__SR0 (happ (h4/semi__ring/semi__ring__SR0__fupd f) s) = happ f (h4/semi__ring/semi__ring__SR0 s)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c0]: !s g f. happ (h4/semi__ring/semi__ring__SR0__fupd f) (happ (h4/semi__ring/semi__ring__SR0__fupd g) s) = happ (h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g)) s
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_combins_ou_u_ASSOC]: !h g f. h4/combin/o f (h4/combin/o g h) = h4/combin/o (h4/combin/o f g) h
% Assm [h4s_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c0]: !g f. h4/combin/o (h4/semi__ring/semi__ring__SR0__fupd f) (h4/semi__ring/semi__ring__SR0__fupd g) = h4/semi__ring/semi__ring__SR0__fupd (h4/combin/o f g)
% Assm [h4s_arithmetics_FUNPOW0u_c1]: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (happ f x)
% Assm [h4s_arithmetics_FUNPOW0u_c0]: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm [h4s_arithmetics_FUNPOWu_u_0]: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_RELu_u_UNIQUE]: !x m2 m1 g2 g1 f. h4/prim__rec/SIMP__REC__REL g1 x f m1 /\ h4/prim__rec/SIMP__REC__REL g2 x f m2 ==> (!n. h4/prim__rec/_3C n m1 /\ h4/prim__rec/_3C n m2 ==> happ g1 n = happ g2 n)
% Assm [h4s_whiles_OWHILEu_u_WHILE]: !s_27 s f G. h4/while/OWHILE G f s = h4/option/SOME s_27 ==> h4/while/WHILE G f s = s_27
% Assm [h4s_relations_IDEM0]: !f. h4/relation/IDEM f <=> (!x. happ f (happ f x) = happ f x)
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_relations_INVOL0]: !f. h4/relation/INVOL f <=> (!x. happ f (happ f x) = x)
% Assm [h4s_arithmetics_FUNPOWu_u_SUC]: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = happ f (h4/arithmetic/FUNPOW f n x)
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_whiles_OWHILEu_u_ENDCOND]: !s_27 s f G. h4/while/OWHILE G f s = h4/option/SOME s_27 ==> ~happ G s_27
% Assm [h4s_relations_INVOLu_u_ONEu_u_ENO]: !f. h4/relation/INVOL f ==> (!a b. happ f a = b <=> a = happ f b)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_relations_IDEMu_u_DEF]: !f. h4/relation/IDEM f <=> h4/combin/o f f = f
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_REL0]: !x n fun f. h4/prim__rec/SIMP__REC__REL fun x f n <=> happ fun h4/num/0 = x /\ (!m. h4/prim__rec/_3C m n ==> happ fun (h4/num/SUC m) = happ f (happ fun m))
% Assm [h4s_primu_u_recs_SUCu_u_LESS]: !n m. h4/prim__rec/_3C (h4/num/SUC m) n ==> h4/prim__rec/_3C m n
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_relations_INVOLu_u_DEF]: !f. h4/relation/INVOL f <=> h4/combin/o f f = h4/combin/I
% Assm [h4s_whiles_OWHILEu_u_def]: !_0. (!G f s n. happ (happ (happ (happ _0 G) f) s) n <=> ~happ G (h4/arithmetic/FUNPOW f n s)) ==> (!s f G. ?v. (v <=> (?n. ~happ G (h4/arithmetic/FUNPOW f n s))) /\ h4/while/OWHILE G f s = h4/bool/COND v (h4/option/SOME (h4/arithmetic/FUNPOW f (h4/while/LEAST (happ (happ (happ _0 G) f) s)) s)) h4/option/NONE)
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c2]: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_whiles_LEASTu_u_ELIM]: !Q P. (?n. happ P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~happ P m) /\ happ P n ==> happ Q n) ==> happ Q (h4/while/LEAST P)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm [h4s_whiles_WHILE0]: !x g P. h4/while/WHILE P g x = h4/bool/COND (happ P x) (h4/while/WHILE P g (happ g x)) x
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% 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_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Goal: !s g f. happ (h4/semi__ring/semi__ring__SRM__fupd f) (happ (h4/semi__ring/semi__ring__SR1__fupd g) s) = happ (h4/semi__ring/semi__ring__SR1__fupd g) (happ (h4/semi__ring/semi__ring__SRM__fupd f) s)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1089145,TV_Q1089141]: ![V_f, V_g]: (![V_x]: s(TV_Q1089141,happ(s(t_fun(TV_Q1089145,TV_Q1089141),V_f),s(TV_Q1089145,V_x))) = s(TV_Q1089141,happ(s(t_fun(TV_Q1089145,TV_Q1089141),V_g),s(TV_Q1089145,V_x))) => s(t_fun(TV_Q1089145,TV_Q1089141),V_f) = s(t_fun(TV_Q1089145,TV_Q1089141),V_g))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27semiu_u_ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r)))).
fof(ah4s_semiu_u_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesemiu_u_u_u_ring0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_x, V_xi_, V_x0, V_xi_0]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring0),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_xi_),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))))))).
fof(ah4s_semiu_u_rings_semiu_u_ring1, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring0)).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_a))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_a)).
fof(ah4s_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_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c3, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c1, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f1),s(TV_u_27a,V_a0))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SRMu_u_fupd0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SR1u_u_fupd0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f1),s(TV_u_27a,V_a0))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c3, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c8, axiom, ![TV_u_27a]: ![V_s, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c3, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c5, axiom, ![TV_u_27a]: ![V_s, V_f]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c2, axiom, ![TV_u_27a]: ![V_s, V_f]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c15, axiom, ![TV_u_27a]: ![V_s, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c2, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c2, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fnu_u_updatesu_c0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f1),s(TV_u_27a,V_a))))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c7, axiom, ![TV_u_27a]: ![V_s, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c13, axiom, ![TV_u_27a]: ![V_s, V_f]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c10, axiom, ![TV_u_27a]: ![V_s, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c1, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c0, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c0, axiom, ![TV_u_27a]: ![V_s, V_f]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a0)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SRPu_u_fupd0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c6, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g)))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c2, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c7, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))).
fof(ah4s_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_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_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_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c1, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),V_h)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c1, axiom, ![TV_u_27a]: ![V_s, V_f]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c14, axiom, ![TV_u_27a]: ![V_s, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c2, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c11, axiom, ![TV_u_27a]: ![V_s, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c4, axiom, ![TV_u_27a]: ![V_s, V_f]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SR0u_u_fupd0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f1))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f1),s(TV_u_27a,V_a))))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c4, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))) <=> ![V_uu_27semiu_u_ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_1))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_rep))))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c3, axiom, ![TV_u_27a]: ![V_s, V_f]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c6, axiom, ![TV_u_27a]: ![V_s, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c9, axiom, ![TV_u_27a]: ![V_s, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accfupdsu_c12, axiom, ![TV_u_27a]: ![V_s, V_f]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_c0, axiom, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))).
fof(ah4s_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_combins_ou_u_ASSOC, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27d,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27d,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(t_fun(TV_u_27d,TV_u_27c),V_h))))) = s(t_fun(TV_u_27d,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(t_fun(TV_u_27d,TV_u_27c),V_h)))).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_fupdfupdsu_u_compu_c0, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_combins_o(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27a,TV_u_27a),V_g)))))).
fof(ah4s_arithmetics_FUNPOW0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_arithmetics_FUNPOW0u_c0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_arithmetics_FUNPOWu_u_0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_primu_u_recs_SIMPu_u_RECu_u_RELu_u_UNIQUE, axiom, ![TV_u_27a]: ![V_x, V_m2, V_m1, V_g2, V_g1, V_f]: ((p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g1),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_m1)))) & p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g2),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_m2))))) => ![V_n]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m1)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m2))))) => s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g1),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g2),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_whiles_OWHILEu_u_WHILE, axiom, ![TV_u_27a]: ![V_su_27, V_s, V_f, V_G]: (s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_su_27))) => s(TV_u_27a,h4s_whiles_while(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(TV_u_27a,V_su_27))).
fof(ah4s_relations_IDEM0, axiom, ![TV_u_27z]: ![V_f]: (p(s(t_bool,h4s_relations_idem(s(t_fun(TV_u_27z,TV_u_27z),V_f)))) <=> ![V_x]: s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,V_x))))) = s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,V_x))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_relations_INVOL0, axiom, ![TV_u_27z]: ![V_f]: (p(s(t_bool,h4s_relations_invol(s(t_fun(TV_u_27z,TV_u_27z),V_f)))) <=> ![V_x]: s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,V_x))))) = s(TV_u_27z,V_x))).
fof(ah4s_arithmetics_FUNPOWu_u_SUC, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x)))))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_whiles_OWHILEu_u_ENDCOND, axiom, ![TV_u_27a]: ![V_su_27, V_s, V_f, V_G]: (s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_su_27))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,V_su_27))))))).
fof(ah4s_relations_INVOLu_u_ONEu_u_ENO, axiom, ![TV_u_27z]: ![V_f]: (p(s(t_bool,h4s_relations_invol(s(t_fun(TV_u_27z,TV_u_27z),V_f)))) => ![V_a, V_b]: (s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,V_a))) = s(TV_u_27z,V_b) <=> s(TV_u_27z,V_a) = s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,V_b)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_relations_IDEMu_u_DEF, axiom, ![TV_u_27z]: ![V_f]: (p(s(t_bool,h4s_relations_idem(s(t_fun(TV_u_27z,TV_u_27z),V_f)))) <=> s(t_fun(TV_u_27z,TV_u_27z),h4s_combins_o(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(t_fun(TV_u_27z,TV_u_27z),V_f))) = s(t_fun(TV_u_27z,TV_u_27z),V_f))).
fof(ah4s_primu_u_recs_SIMPu_u_RECu_u_REL0, axiom, ![TV_u_27a]: ![V_x, V_n, V_fun, V_f]: (p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))) <=> (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_x) & ![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,V_m))))))))).
fof(ah4s_primu_u_recs_SUCu_u_LESS, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
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,f0))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_relations_INVOLu_u_DEF, axiom, ![TV_u_27z]: ![V_f]: (p(s(t_bool,h4s_relations_invol(s(t_fun(TV_u_27z,TV_u_27z),V_f)))) <=> s(t_fun(TV_u_27z,TV_u_27z),h4s_combins_o(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(t_fun(TV_u_27z,TV_u_27z),V_f))) = s(t_fun(TV_u_27z,TV_u_27z),h4s_combins_i))).
fof(ah4s_whiles_OWHILEu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_G, V_f, V_s, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_G))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(TV_u_27a,V_s))),s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_s)))))))) => ![V_s, V_f, V_G]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_n]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_s)))))))) & s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_G))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(TV_u_27a,V_s))))),s(TV_u_27a,V_s))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> (p(s(t_bool,V_P)) & s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_whiles_LEASTu_u_ELIM, axiom, ![V_Q, V_P]: ((?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_n]: ((![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,V_n)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_whiles_WHILE0, axiom, ![TV_u_27a]: ![V_x, V_g, V_P]: s(TV_u_27a,h4s_whiles_while(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,TV_u_27a),V_g),s(TV_u_27a,V_x))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))),s(TV_u_27a,h4s_whiles_while(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,TV_u_27a),V_g),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
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_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
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,f0))))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) & ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_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_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_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_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_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_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_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_dcu_u_conj, 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_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
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_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_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,f0))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | 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,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
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,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ch4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_c4, conjecture, ![TV_u_27a]: ![V_s, V_g, V_f]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_s)))))).
