%   ORIGINAL: h4/ring/ring__fupdcanon__comp_c12
% 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/ring/hidden____20__40ind____typering0____def: h4/ring/_20_40ind__typering0 = (\a0 a1 a2 a3 a4. h4/ring/_20_40ind__typering1 ((\a00 a10 a20 a30 a40. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 (h4/pair/_2C a30 a40)))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3 a4))
% Assm: h4/ring/ring__repfns_c1: !r. (\a0_27. !_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = (\a00 a10 a20 a30 a40. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 (h4/pair/_2C a30 a40)))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3 a4) ==> _27ring_27 a0_270) ==> _27ring_27 a0_27) r <=> h4/ring/_20_40ind__typering2 (h4/ring/_20_40ind__typering1 r) = r
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/ring/ring1: h4/ring/ring0 = h4/ring/_20_40ind__typering0
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/TRUTH: T
% Assm: h4/ring/ring__repfns_c0: !a. h4/ring/_20_40ind__typering1 (h4/ring/_20_40ind__typering2 a) = a
% Assm: h4/ring/ring__fupdfupds__comp_c8: !g f. h4/combin/o (h4/ring/ring__RN__fupd f) (h4/ring/ring__RN__fupd g) = h4/ring/ring__RN__fupd (h4/combin/o f g)
% Assm: h4/ring/ring__fn__updates_c4: !f2 f1 f0 f a0 a. h4/ring/ring__RN__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 f f0 (f2 f1)
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/ring/ring__fn__updates_c0: !f2 f1 f0 f a0 a. h4/ring/ring__R0__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 (f2 a) a0 f f0 f1
% Assm: h4/ring/ring__RN__fupd0: !f2 f1 f0 f a0 a. h4/ring/ring__RN__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 f f0 (f2 f1)
% Assm: h4/ring/ring__fupdcanon__comp_c0: !g f. h4/combin/o (h4/ring/ring__R1__fupd f) (h4/ring/ring__R0__fupd g) = h4/combin/o (h4/ring/ring__R0__fupd g) (h4/ring/ring__R1__fupd f)
% Assm: h4/ring/ring__fupdcanon__comp_c6: !g f. h4/combin/o (h4/ring/ring__RM__fupd f) (h4/ring/ring__R0__fupd g) = h4/combin/o (h4/ring/ring__R0__fupd g) (h4/ring/ring__RM__fupd f)
% Assm: h4/ring/ring__fupdfupds__comp_c0: !g f. h4/combin/o (h4/ring/ring__R0__fupd f) (h4/ring/ring__R0__fupd g) = h4/ring/ring__R0__fupd (h4/combin/o f g)
% Assm: h4/ring/ring__fupdcanon__comp_c2: !g f. h4/combin/o (h4/ring/ring__RP__fupd f) (h4/ring/ring__R0__fupd g) = h4/combin/o (h4/ring/ring__R0__fupd g) (h4/ring/ring__RP__fupd f)
% Assm: h4/ring/ring__fupdcanon_c6: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__RN__fupd f r)
% Assm: h4/ring/ring__R0__fupd0: !f2 f1 f0 f a0 a. h4/ring/ring__R0__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 (f2 a) a0 f f0 f1
% Assm: h4/ring/ring__fn__updates_c1: !f2 f1 f0 f a0 a. h4/ring/ring__R1__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a (f2 a0) f f0 f1
% Assm: h4/ring/ring__fn__updates_c3: !f2 f1 f0 f a0 a. h4/ring/ring__RM__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 f (f2 f0) f1
% Assm: h4/ring/ring__fn__updates_c2: !f2 f1 f0 f a0 a. h4/ring/ring__RP__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 (f2 f) f0 f1
% Assm: h4/ring/ring__fupdfupds_c4: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RN__fupd g r) = h4/ring/ring__RN__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__fupdcanon_c7: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RN__fupd f r)
% Assm: h4/ring/ring__accfupds_c11: !r f. h4/ring/ring__RP (h4/ring/ring__RN__fupd f r) = h4/ring/ring__RP r
% Assm: h4/ring/ring__fupdcanon_c9: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RM__fupd g r) = h4/ring/ring__RM__fupd g (h4/ring/ring__RN__fupd f r)
% Assm: h4/ring/ring__accfupds_c3: !r f. h4/ring/ring__R0 (h4/ring/ring__RN__fupd f r) = h4/ring/ring__R0 r
% Assm: h4/ring/ring__accfupds_c24: !r f. h4/ring/ring__RN (h4/ring/ring__RN__fupd f r) = f (h4/ring/ring__RN r)
% Assm: h4/ring/ring__accfupds_c7: !r f. h4/ring/ring__R1 (h4/ring/ring__RN__fupd f r) = h4/ring/ring__R1 r
% Assm: h4/ring/ring__accfupds_c15: !r f. h4/ring/ring__RM (h4/ring/ring__RN__fupd f r) = h4/ring/ring__RM r
% Assm: h4/ring/ring__fupdcanon_c8: !r g f. h4/ring/ring__RN__fupd f (h4/ring/ring__RP__fupd g r) = h4/ring/ring__RP__fupd g (h4/ring/ring__RN__fupd f r)
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ring/ring__fupdfupds__comp_c9: !h g f. h4/combin/o (h4/ring/ring__RN__fupd f) (h4/combin/o (h4/ring/ring__RN__fupd g) h) = h4/combin/o (h4/ring/ring__RN__fupd (h4/combin/o f g)) h
% 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/ring/ring__accessors_c2: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm: h4/ring/ring__accessors_c0: !f1 f0 f a0 a. h4/ring/ring__R0 (h4/ring/ring0 a a0 f f0 f1) = a
% Assm: h4/ring/ring__accessors_c4: !f1 f0 f a0 a. h4/ring/ring__RN (h4/ring/ring0 a a0 f f0 f1) = f1
% Assm: h4/ring/ring__accessors_c1: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm: h4/ring/ring__accessors_c3: !f1 f0 f a0 a. h4/ring/ring__RM (h4/ring/ring0 a a0 f f0 f1) = f0
% Assm: h4/ring/ring__fupdfupds_c0: !r g f. h4/ring/ring__R0__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__fupdcanon__comp_c1: !h g f. h4/combin/o (h4/ring/ring__R1__fupd f) (h4/combin/o (h4/ring/ring__R0__fupd g) h) = h4/combin/o (h4/ring/ring__R0__fupd g) (h4/combin/o (h4/ring/ring__R1__fupd f) h)
% Assm: h4/ring/ring__fupdcanon__comp_c3: !h g f. h4/combin/o (h4/ring/ring__RP__fupd f) (h4/combin/o (h4/ring/ring__R0__fupd g) h) = h4/combin/o (h4/ring/ring__R0__fupd g) (h4/combin/o (h4/ring/ring__RP__fupd f) h)
% Assm: h4/ring/ring__fupdcanon__comp_c7: !h g f. h4/combin/o (h4/ring/ring__RM__fupd f) (h4/combin/o (h4/ring/ring__R0__fupd g) h) = h4/combin/o (h4/ring/ring__R0__fupd g) (h4/combin/o (h4/ring/ring__RM__fupd f) h)
% Assm: h4/ring/ring__fupdfupds__comp_c1: !h g f. h4/combin/o (h4/ring/ring__R0__fupd f) (h4/combin/o (h4/ring/ring__R0__fupd g) h) = h4/combin/o (h4/ring/ring__R0__fupd (h4/combin/o f g)) h
% Assm: h4/ring/ring__accfupds_c16: !r f. h4/ring/ring__RN (h4/ring/ring__R0__fupd f r) = h4/ring/ring__RN r
% Assm: h4/ring/ring__accfupds_c8: !r f. h4/ring/ring__RP (h4/ring/ring__R0__fupd f r) = h4/ring/ring__RP r
% Assm: h4/ring/ring__accfupds_c4: !r f. h4/ring/ring__R1 (h4/ring/ring__R0__fupd f r) = h4/ring/ring__R1 r
% Assm: h4/ring/ring__fupdcanon_c3: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__RM__fupd f r)
% Assm: h4/ring/ring__accfupds_c20: !r f. h4/ring/ring__R0 (h4/ring/ring__R0__fupd f r) = f (h4/ring/ring__R0 r)
% Assm: h4/ring/ring__accfupds_c12: !r f. h4/ring/ring__RM (h4/ring/ring__R0__fupd f r) = h4/ring/ring__RM r
% Assm: h4/ring/ring__fupdcanon_c1: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__RP__fupd f r)
% Assm: h4/ring/ring__fupdcanon_c0: !r g f. h4/ring/ring__R1__fupd f (h4/ring/ring__R0__fupd g r) = h4/ring/ring__R0__fupd g (h4/ring/ring__R1__fupd f r)
% Assm: h4/ring/ring__RM0: !f1 f0 f a0 a. h4/ring/ring__RM (h4/ring/ring0 a a0 f f0 f1) = f0
% Assm: h4/ring/ring__R1__fupd0: !f2 f1 f0 f a0 a. h4/ring/ring__R1__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a (f2 a0) f f0 f1
% Assm: h4/ring/ring__R10: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm: h4/ring/ring__RM__fupd0: !f2 f1 f0 f a0 a. h4/ring/ring__RM__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 f (f2 f0) f1
% Assm: h4/ring/ring__RP0: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm: h4/ring/ring__RP__fupd0: !f2 f1 f0 f a0 a. h4/ring/ring__RP__fupd f2 (h4/ring/ring0 a a0 f f0 f1) = h4/ring/ring0 a a0 (f2 f) f0 f1
% Assm: h4/ring/ring__R00: !f1 f0 f a0 a. h4/ring/ring__R0 (h4/ring/ring0 a a0 f f0 f1) = a
% Assm: h4/ring/ring__RN0: !f1 f0 f a0 a. h4/ring/ring__RN (h4/ring/ring0 a a0 f f0 f1) = f1
% Assm: h4/ring/ring__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = (\a00 a10 a20 a30 a40. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 (h4/pair/_2C a30 a40)))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3 a4) ==> _27ring_27 a0_270) ==> _27ring_27 a0_27) rep
% Assm: h4/ring/ring__fupdfupds__comp_c2: !g f. h4/combin/o (h4/ring/ring__R1__fupd f) (h4/ring/ring__R1__fupd g) = h4/ring/ring__R1__fupd (h4/combin/o f g)
% Assm: h4/ring/ring__fupdcanon__comp_c4: !g f. h4/combin/o (h4/ring/ring__RP__fupd f) (h4/ring/ring__R1__fupd g) = h4/combin/o (h4/ring/ring__R1__fupd g) (h4/ring/ring__RP__fupd f)
% Assm: h4/ring/ring__fupdfupds_c1: !r g f. h4/ring/ring__R1__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__fupdcanon__comp_c8: !g f. h4/combin/o (h4/ring/ring__RM__fupd f) (h4/ring/ring__R1__fupd g) = h4/combin/o (h4/ring/ring__R1__fupd g) (h4/ring/ring__RM__fupd f)
% Assm: h4/ring/ring__fupdcanon__comp_c5: !h g f. h4/combin/o (h4/ring/ring__RP__fupd f) (h4/combin/o (h4/ring/ring__R1__fupd g) h) = h4/combin/o (h4/ring/ring__R1__fupd g) (h4/combin/o (h4/ring/ring__RP__fupd f) h)
% Assm: h4/ring/ring__fupdcanon__comp_c9: !h g f. h4/combin/o (h4/ring/ring__RM__fupd f) (h4/combin/o (h4/ring/ring__R1__fupd g) h) = h4/combin/o (h4/ring/ring__R1__fupd g) (h4/combin/o (h4/ring/ring__RM__fupd f) h)
% Assm: h4/ring/ring__fupdfupds__comp_c3: !h g f. h4/combin/o (h4/ring/ring__R1__fupd f) (h4/combin/o (h4/ring/ring__R1__fupd g) h) = h4/combin/o (h4/ring/ring__R1__fupd (h4/combin/o f g)) h
% Assm: h4/ring/ring__fupdfupds__comp_c4: !g f. h4/combin/o (h4/ring/ring__RP__fupd f) (h4/ring/ring__RP__fupd g) = h4/ring/ring__RP__fupd (h4/combin/o f g)
% Assm: h4/ring/ring__fupdfupds__comp_c7: !h g f. h4/combin/o (h4/ring/ring__RM__fupd f) (h4/combin/o (h4/ring/ring__RM__fupd g) h) = h4/combin/o (h4/ring/ring__RM__fupd (h4/combin/o f g)) h
% Assm: h4/ring/ring__fupdfupds_c3: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__RM__fupd g r) = h4/ring/ring__RM__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__fupdfupds_c2: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__RP__fupd g r) = h4/ring/ring__RP__fupd (h4/combin/o f g) r
% Assm: h4/ring/ring__fupdcanon__comp_c10: !g f. h4/combin/o (h4/ring/ring__RM__fupd f) (h4/ring/ring__RP__fupd g) = h4/combin/o (h4/ring/ring__RP__fupd g) (h4/ring/ring__RM__fupd f)
% Assm: h4/ring/ring__fupdcanon__comp_c11: !h g f. h4/combin/o (h4/ring/ring__RM__fupd f) (h4/combin/o (h4/ring/ring__RP__fupd g) h) = h4/combin/o (h4/ring/ring__RP__fupd g) (h4/combin/o (h4/ring/ring__RM__fupd f) h)
% Assm: h4/ring/ring__fupdfupds__comp_c5: !h g f. h4/combin/o (h4/ring/ring__RP__fupd f) (h4/combin/o (h4/ring/ring__RP__fupd g) h) = h4/combin/o (h4/ring/ring__RP__fupd (h4/combin/o f g)) h
% Assm: h4/ring/ring__fupdfupds__comp_c6: !g f. h4/combin/o (h4/ring/ring__RM__fupd f) (h4/ring/ring__RM__fupd g) = h4/ring/ring__RM__fupd (h4/combin/o f g)
% Assm: h4/ring/ring__size__def: !f a4 a3 a2 a1 a0. h4/ring/ring__size f (h4/ring/ring0 a0 a1 a2 a3 a4) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (f a1))
% Assm: h4/ring/ring__accfupds_c13: !r f. h4/ring/ring__RM (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RM r
% Assm: h4/ring/ring__accfupds_c21: !r f. h4/ring/ring__R1 (h4/ring/ring__R1__fupd f r) = f (h4/ring/ring__R1 r)
% 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/ring/ring__accfupds_c9: !r f. h4/ring/ring__RP (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RP r
% Assm: h4/ring/ring__accfupds_c0: !r f. h4/ring/ring__R0 (h4/ring/ring__R1__fupd f r) = h4/ring/ring__R0 r
% Assm: h4/ring/ring__fupdcanon_c2: !r g f. h4/ring/ring__RP__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RP__fupd f r)
% Assm: h4/ring/ring__accfupds_c17: !r f. h4/ring/ring__RN (h4/ring/ring__R1__fupd f r) = h4/ring/ring__RN r
% Assm: h4/ring/ring__fupdcanon_c4: !r g f. h4/ring/ring__RM__fupd f (h4/ring/ring__R1__fupd g r) = h4/ring/ring__R1__fupd g (h4/ring/ring__RM__fupd f r)
% Assm: h4/list/LIST__APPLY__o: !xs gs fs. h4/list/LIST__APPLY (h4/list/LIST__APPLY (h4/list/LIST__APPLY (h4/list/CONS h4/combin/o h4/list/NIL) fs) gs) xs = h4/list/LIST__APPLY fs (h4/list/LIST__APPLY gs xs)
% Assm: h4/option/OPTION__APPLY__o: !x g f. h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/SOME h4/combin/o) f) g) x = h4/option/OPTION__APPLY f (h4/option/OPTION__APPLY g x)
% Assm: h4/relation/INVOL__DEF: !f. h4/relation/INVOL f <=> h4/combin/o f f = h4/combin/I
% Assm: h4/semi__ring/semi__ring__fupdcanon__comp_c8: !g f. h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f) (h4/semi__ring/semi__ring__SR1__fupd g) = h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g) (h4/semi__ring/semi__ring__SRM__fupd f)
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/OPTION__APPLY__def_c1: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__APPLY__def_c0: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% 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__repfns_c1: !r. (\a0_27. !_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3) ==> _27semi__ring_27 a0_270) ==> _27semi__ring_27 a0_27) r <=> h4/semi__ring/_20_40ind__typesemi__ring2 (h4/semi__ring/_20_40ind__typesemi__ring1 r) = r
% Assm: h4/semi__ring/semi__ring__repfns_c0: !a. h4/semi__ring/_20_40ind__typesemi__ring1 (h4/semi__ring/_20_40ind__typesemi__ring2 a) = a
% 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/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/list/LIST__APPLY__DEF: !xs fs. h4/list/LIST__APPLY fs xs = h4/list/LIST__BIND fs (h4/combin/C h4/list/MAP xs)
% Assm: h4/list/SINGL__LIST__APPLY__L: !x f. h4/list/LIST__BIND (h4/list/CONS x h4/list/NIL) f = f x
% Assm: h4/list/LIST__BIND__APPEND: !l2 l1 f. h4/list/LIST__BIND (h4/list/APPEND l1 l2) f = h4/list/APPEND (h4/list/LIST__BIND l1 f) (h4/list/LIST__BIND l2 f)
% Assm: h4/list/LIST__BIND__MAP: !l g f. h4/list/LIST__BIND (h4/list/MAP f l) g = h4/list/LIST__BIND l (h4/combin/o g f)
% Assm: h4/list/LIST__BIND__THM_c1: !t h f. h4/list/LIST__BIND (h4/list/CONS h t) f = h4/list/APPEND (f h) (h4/list/LIST__BIND t f)
% Assm: h4/list/MAP__LIST__BIND: !l g f. h4/list/MAP f (h4/list/LIST__BIND l g) = h4/list/LIST__BIND l (h4/combin/o (h4/list/MAP f) g)
% Assm: h4/list/LIST__BIND__THM_c0: !f. h4/list/LIST__BIND h4/list/NIL f = h4/list/NIL
% Assm: h4/list/APPEND__11_c1: !l3 l2 l1. h4/list/APPEND l2 l1 = h4/list/APPEND l3 l1 <=> l2 = l3
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/list__repfns_c1: !r. (\a0_27. !_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_27 a1) ==> _27list_27 a0_270) ==> _27list_27 a0_27) r <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/hidden____20__40ind____typelist0____def: h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/list/MAP__MAP__o: !l g f. h4/list/MAP f (h4/list/MAP g l) = h4/list/MAP (h4/combin/o f g) l
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Goal: !g f. h4/combin/o (h4/ring/ring__RN__fupd f) (h4/ring/ring__R0__fupd g) = h4/combin/o (h4/ring/ring__R0__fupd g) (h4/ring/ring__RN__fupd f)
%   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_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typering0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x' x x' x. happ (happ (happ (happ (happ h4/ring/_20_40ind__typering0 x) x') x) x') x = h4/ring/_20_40ind__typering1 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x (h4/pair/_2C x' (h4/pair/_2C x (h4/pair/_2C x' x)))) _0))
% Assm [h4s_rings_ringu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 (h4/pair/_2C a3 a4)))) _0) ==> happ _27ring_27 a0_270) ==> happ _27ring_27 r) <=> h4/ring/_20_40ind__typering2 (h4/ring/_20_40ind__typering1 r) = r)
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_rings_ring1]: h4/ring/ring0 = h4/ring/_20_40ind__typering0
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_rings_ringu_u_repfnsu_c0]: !a. h4/ring/_20_40ind__typering1 (h4/ring/_20_40ind__typering2 a) = a
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c8]: !g f. happ (happ h4/combin/o (h4/ring/ring__RN__fupd f)) (h4/ring/ring__RN__fupd g) = h4/ring/ring__RN__fupd (happ (happ h4/combin/o f) g)
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c4]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__RN__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) (happ f2 f1)
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (happ (happ h4/combin/o f) g) x = happ f (happ g x)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c0]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__R0__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 (happ f2 a)) a0) f) f0) f1
% Assm [h4s_rings_ringu_u_RNu_u_fupd0]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__RN__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) (happ f2 f1)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c0]: !g f. happ (happ h4/combin/o (h4/ring/ring__R1__fupd f)) (h4/ring/ring__R0__fupd g) = happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) (h4/ring/ring__R1__fupd f)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c6]: !g f. happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) (h4/ring/ring__R0__fupd g) = happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) (h4/ring/ring__RM__fupd f)
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c0]: !g f. happ (happ h4/combin/o (h4/ring/ring__R0__fupd f)) (h4/ring/ring__R0__fupd g) = h4/ring/ring__R0__fupd (happ (happ h4/combin/o f) g)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c2]: !g f. happ (happ h4/combin/o (h4/ring/ring__RP__fupd f)) (h4/ring/ring__R0__fupd g) = happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) (h4/ring/ring__RP__fupd f)
% Assm [h4s_rings_ringu_u_fupdcanonu_c6]: !r g f. happ (h4/ring/ring__RN__fupd f) (happ (h4/ring/ring__R0__fupd g) r) = happ (h4/ring/ring__R0__fupd g) (happ (h4/ring/ring__RN__fupd f) r)
% Assm [h4s_rings_ringu_u_R0u_u_fupd0]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__R0__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 (happ f2 a)) a0) f) f0) f1
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c1]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__R1__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) (happ f2 a0)) f) f0) f1
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c3]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__RM__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) (happ f2 f0)) f1
% Assm [h4s_rings_ringu_u_fnu_u_updatesu_c2]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__RP__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) (happ f2 f)) f0) f1
% Assm [h4s_rings_ringu_u_fupdfupdsu_c4]: !r g f. happ (h4/ring/ring__RN__fupd f) (happ (h4/ring/ring__RN__fupd g) r) = happ (h4/ring/ring__RN__fupd (happ (happ h4/combin/o f) g)) r
% Assm [h4s_rings_ringu_u_fupdcanonu_c7]: !r g f. happ (h4/ring/ring__RN__fupd f) (happ (h4/ring/ring__R1__fupd g) r) = happ (h4/ring/ring__R1__fupd g) (happ (h4/ring/ring__RN__fupd f) r)
% Assm [h4s_rings_ringu_u_accfupdsu_c11]: !r f. h4/ring/ring__RP (happ (h4/ring/ring__RN__fupd f) r) = h4/ring/ring__RP r
% Assm [h4s_rings_ringu_u_fupdcanonu_c9]: !r g f. happ (h4/ring/ring__RN__fupd f) (happ (h4/ring/ring__RM__fupd g) r) = happ (h4/ring/ring__RM__fupd g) (happ (h4/ring/ring__RN__fupd f) r)
% Assm [h4s_rings_ringu_u_accfupdsu_c3]: !r f. h4/ring/ring__R0 (happ (h4/ring/ring__RN__fupd f) r) = h4/ring/ring__R0 r
% Assm [h4s_rings_ringu_u_accfupdsu_c24]: !r f. h4/ring/ring__RN (happ (h4/ring/ring__RN__fupd f) r) = happ f (h4/ring/ring__RN r)
% Assm [h4s_rings_ringu_u_accfupdsu_c7]: !r f. h4/ring/ring__R1 (happ (h4/ring/ring__RN__fupd f) r) = h4/ring/ring__R1 r
% Assm [h4s_rings_ringu_u_accfupdsu_c15]: !r f. h4/ring/ring__RM (happ (h4/ring/ring__RN__fupd f) r) = h4/ring/ring__RM r
% Assm [h4s_rings_ringu_u_fupdcanonu_c8]: !r g f. happ (h4/ring/ring__RN__fupd f) (happ (h4/ring/ring__RP__fupd g) r) = happ (h4/ring/ring__RP__fupd g) (happ (h4/ring/ring__RN__fupd f) r)
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c9]: !h g f. happ (happ h4/combin/o (h4/ring/ring__RN__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__RN__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__RN__fupd (happ (happ h4/combin/o f) g))) h
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_combins_ou_u_ASSOC]: !h g f. happ (happ h4/combin/o f) (happ (happ h4/combin/o g) h) = happ (happ h4/combin/o (happ (happ h4/combin/o f) g)) h
% Assm [h4s_rings_ringu_u_accessorsu_c2]: !f1 f0 f a0 a. h4/ring/ring__RP (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f
% Assm [h4s_rings_ringu_u_accessorsu_c0]: !f1 f0 f a0 a. h4/ring/ring__R0 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = a
% Assm [h4s_rings_ringu_u_accessorsu_c4]: !f1 f0 f a0 a. h4/ring/ring__RN (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f1
% Assm [h4s_rings_ringu_u_accessorsu_c1]: !f1 f0 f a0 a. h4/ring/ring__R1 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = a0
% Assm [h4s_rings_ringu_u_accessorsu_c3]: !f1 f0 f a0 a. h4/ring/ring__RM (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f0
% Assm [h4s_rings_ringu_u_fupdfupdsu_c0]: !r g f. happ (h4/ring/ring__R0__fupd f) (happ (h4/ring/ring__R0__fupd g) r) = happ (h4/ring/ring__R0__fupd (happ (happ h4/combin/o f) g)) r
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c1]: !h g f. happ (happ h4/combin/o (h4/ring/ring__R1__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) (happ (happ h4/combin/o (h4/ring/ring__R1__fupd f)) h)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c3]: !h g f. happ (happ h4/combin/o (h4/ring/ring__RP__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) (happ (happ h4/combin/o (h4/ring/ring__RP__fupd f)) h)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c7]: !h g f. happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) (happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) h)
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c1]: !h g f. happ (happ h4/combin/o (h4/ring/ring__R0__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__R0__fupd (happ (happ h4/combin/o f) g))) h
% Assm [h4s_rings_ringu_u_accfupdsu_c16]: !r f. h4/ring/ring__RN (happ (h4/ring/ring__R0__fupd f) r) = h4/ring/ring__RN r
% Assm [h4s_rings_ringu_u_accfupdsu_c8]: !r f. h4/ring/ring__RP (happ (h4/ring/ring__R0__fupd f) r) = h4/ring/ring__RP r
% Assm [h4s_rings_ringu_u_accfupdsu_c4]: !r f. h4/ring/ring__R1 (happ (h4/ring/ring__R0__fupd f) r) = h4/ring/ring__R1 r
% Assm [h4s_rings_ringu_u_fupdcanonu_c3]: !r g f. happ (h4/ring/ring__RM__fupd f) (happ (h4/ring/ring__R0__fupd g) r) = happ (h4/ring/ring__R0__fupd g) (happ (h4/ring/ring__RM__fupd f) r)
% Assm [h4s_rings_ringu_u_accfupdsu_c20]: !r f. h4/ring/ring__R0 (happ (h4/ring/ring__R0__fupd f) r) = happ f (h4/ring/ring__R0 r)
% Assm [h4s_rings_ringu_u_accfupdsu_c12]: !r f. h4/ring/ring__RM (happ (h4/ring/ring__R0__fupd f) r) = h4/ring/ring__RM r
% Assm [h4s_rings_ringu_u_fupdcanonu_c1]: !r g f. happ (h4/ring/ring__RP__fupd f) (happ (h4/ring/ring__R0__fupd g) r) = happ (h4/ring/ring__R0__fupd g) (happ (h4/ring/ring__RP__fupd f) r)
% Assm [h4s_rings_ringu_u_fupdcanonu_c0]: !r g f. happ (h4/ring/ring__R1__fupd f) (happ (h4/ring/ring__R0__fupd g) r) = happ (h4/ring/ring__R0__fupd g) (happ (h4/ring/ring__R1__fupd f) r)
% Assm [h4s_rings_ringu_u_RM0]: !f1 f0 f a0 a. h4/ring/ring__RM (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f0
% Assm [h4s_rings_ringu_u_R1u_u_fupd0]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__R1__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) (happ f2 a0)) f) f0) f1
% Assm [h4s_rings_ringu_u_R10]: !f1 f0 f a0 a. h4/ring/ring__R1 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = a0
% Assm [h4s_rings_ringu_u_RMu_u_fupd0]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__RM__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) (happ f2 f0)) f1
% Assm [h4s_rings_ringu_u_RP0]: !f1 f0 f a0 a. h4/ring/ring__RP (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f
% Assm [h4s_rings_ringu_u_RPu_u_fupd0]: !f2 f1 f0 f a0 a. happ (h4/ring/ring__RP__fupd f2) (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = happ (happ (happ (happ (happ h4/ring/ring0 a) a0) (happ f2 f)) f0) f1
% Assm [h4s_rings_ringu_u_R00]: !f1 f0 f a0 a. h4/ring/ring__R0 (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = a
% Assm [h4s_rings_ringu_u_RN0]: !f1 f0 f a0 a. h4/ring/ring__RN (happ (happ (happ (happ (happ h4/ring/ring0 a) a0) f) f0) f1) = f1
% Assm [h4s_rings_ringu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27ring_27. (!a0_270. (?a0 a1 a2 a3 a4. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 (h4/pair/_2C a3 a4)))) _1) ==> happ _27ring_27 a0_270) ==> happ _27ring_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c2]: !g f. happ (happ h4/combin/o (h4/ring/ring__R1__fupd f)) (h4/ring/ring__R1__fupd g) = h4/ring/ring__R1__fupd (happ (happ h4/combin/o f) g)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c4]: !g f. happ (happ h4/combin/o (h4/ring/ring__RP__fupd f)) (h4/ring/ring__R1__fupd g) = happ (happ h4/combin/o (h4/ring/ring__R1__fupd g)) (h4/ring/ring__RP__fupd f)
% Assm [h4s_rings_ringu_u_fupdfupdsu_c1]: !r g f. happ (h4/ring/ring__R1__fupd f) (happ (h4/ring/ring__R1__fupd g) r) = happ (h4/ring/ring__R1__fupd (happ (happ h4/combin/o f) g)) r
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c8]: !g f. happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) (h4/ring/ring__R1__fupd g) = happ (happ h4/combin/o (h4/ring/ring__R1__fupd g)) (h4/ring/ring__RM__fupd f)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c5]: !h g f. happ (happ h4/combin/o (h4/ring/ring__RP__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__R1__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__R1__fupd g)) (happ (happ h4/combin/o (h4/ring/ring__RP__fupd f)) h)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c9]: !h g f. happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__R1__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__R1__fupd g)) (happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) h)
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c3]: !h g f. happ (happ h4/combin/o (h4/ring/ring__R1__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__R1__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__R1__fupd (happ (happ h4/combin/o f) g))) h
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c4]: !g f. happ (happ h4/combin/o (h4/ring/ring__RP__fupd f)) (h4/ring/ring__RP__fupd g) = h4/ring/ring__RP__fupd (happ (happ h4/combin/o f) g)
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c7]: !h g f. happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__RM__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__RM__fupd (happ (happ h4/combin/o f) g))) h
% Assm [h4s_rings_ringu_u_fupdfupdsu_c3]: !r g f. happ (h4/ring/ring__RM__fupd f) (happ (h4/ring/ring__RM__fupd g) r) = happ (h4/ring/ring__RM__fupd (happ (happ h4/combin/o f) g)) r
% Assm [h4s_rings_ringu_u_fupdfupdsu_c2]: !r g f. happ (h4/ring/ring__RP__fupd f) (happ (h4/ring/ring__RP__fupd g) r) = happ (h4/ring/ring__RP__fupd (happ (happ h4/combin/o f) g)) r
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c10]: !g f. happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) (h4/ring/ring__RP__fupd g) = happ (happ h4/combin/o (h4/ring/ring__RP__fupd g)) (h4/ring/ring__RM__fupd f)
% Assm [h4s_rings_ringu_u_fupdcanonu_u_compu_c11]: !h g f. happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__RP__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__RP__fupd g)) (happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) h)
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c5]: !h g f. happ (happ h4/combin/o (h4/ring/ring__RP__fupd f)) (happ (happ h4/combin/o (h4/ring/ring__RP__fupd g)) h) = happ (happ h4/combin/o (h4/ring/ring__RP__fupd (happ (happ h4/combin/o f) g))) h
% Assm [h4s_rings_ringu_u_fupdfupdsu_u_compu_c6]: !g f. happ (happ h4/combin/o (h4/ring/ring__RM__fupd f)) (h4/ring/ring__RM__fupd g) = h4/ring/ring__RM__fupd (happ (happ h4/combin/o f) g)
% Assm [h4s_rings_ringu_u_sizeu_u_def]: !f a4 a3 a2 a1 a0. h4/ring/ring__size f (happ (happ (happ (happ (happ h4/ring/ring0 a0) a1) a2) a3) a4) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (happ f a1))
% Assm [h4s_rings_ringu_u_accfupdsu_c13]: !r f. h4/ring/ring__RM (happ (h4/ring/ring__R1__fupd f) r) = h4/ring/ring__RM r
% Assm [h4s_rings_ringu_u_accfupdsu_c21]: !r f. h4/ring/ring__R1 (happ (h4/ring/ring__R1__fupd f) r) = happ f (h4/ring/ring__R1 r)
% 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_rings_ringu_u_accfupdsu_c9]: !r f. h4/ring/ring__RP (happ (h4/ring/ring__R1__fupd f) r) = h4/ring/ring__RP r
% Assm [h4s_rings_ringu_u_accfupdsu_c0]: !r f. h4/ring/ring__R0 (happ (h4/ring/ring__R1__fupd f) r) = h4/ring/ring__R0 r
% Assm [h4s_rings_ringu_u_fupdcanonu_c2]: !r g f. happ (h4/ring/ring__RP__fupd f) (happ (h4/ring/ring__R1__fupd g) r) = happ (h4/ring/ring__R1__fupd g) (happ (h4/ring/ring__RP__fupd f) r)
% Assm [h4s_rings_ringu_u_accfupdsu_c17]: !r f. h4/ring/ring__RN (happ (h4/ring/ring__R1__fupd f) r) = h4/ring/ring__RN r
% Assm [h4s_rings_ringu_u_fupdcanonu_c4]: !r g f. happ (h4/ring/ring__RM__fupd f) (happ (h4/ring/ring__R1__fupd g) r) = happ (h4/ring/ring__R1__fupd g) (happ (h4/ring/ring__RM__fupd f) r)
% Assm [h4s_lists_LISTu_u_APPLYu_u_o]: !xs gs fs. h4/list/LIST__APPLY (h4/list/LIST__APPLY (h4/list/LIST__APPLY (happ (happ h4/list/CONS h4/combin/o) h4/list/NIL) fs) gs) xs = h4/list/LIST__APPLY fs (h4/list/LIST__APPLY gs xs)
% Assm [h4s_options_OPTIONu_u_APPLYu_u_o]: !x g f. h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/SOME h4/combin/o) f) g) x = h4/option/OPTION__APPLY f (h4/option/OPTION__APPLY g x)
% Assm [h4s_relations_INVOLu_u_DEF]: !f. h4/relation/INVOL f <=> happ (happ h4/combin/o f) f = h4/combin/I
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c8]: !g f. happ (happ h4/combin/o (h4/semi__ring/semi__ring__SRM__fupd f)) (h4/semi__ring/semi__ring__SR1__fupd g) = happ (happ h4/combin/o (h4/semi__ring/semi__ring__SR1__fupd g)) (h4/semi__ring/semi__ring__SRM__fupd f)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c1]: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c0]: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% 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_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 a3))) _0) ==> happ _27semi__ring_27 a0_270) ==> happ _27semi__ring_27 r) <=> h4/semi__ring/_20_40ind__typesemi__ring2 (h4/semi__ring/_20_40ind__typesemi__ring1 r) = r)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c0]: !a. h4/semi__ring/_20_40ind__typesemi__ring1 (h4/semi__ring/_20_40ind__typesemi__ring2 a) = a
% Assm [h4s_semiu_u_rings_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_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_lists_LISTu_u_APPLYu_u_DEF]: !xs fs. h4/list/LIST__APPLY fs xs = h4/list/LIST__BIND fs (h4/combin/C h4/list/MAP xs)
% Assm [h4s_lists_SINGLu_u_LISTu_u_APPLYu_u_L]: !x f. h4/list/LIST__BIND (happ (happ h4/list/CONS x) h4/list/NIL) f = happ f x
% Assm [h4s_lists_LISTu_u_BINDu_u_APPEND]: !l2 l1 f. h4/list/LIST__BIND (h4/list/APPEND l1 l2) f = h4/list/APPEND (h4/list/LIST__BIND l1 f) (h4/list/LIST__BIND l2 f)
% Assm [h4s_lists_LISTu_u_BINDu_u_MAP]: !l g f. h4/list/LIST__BIND (happ (happ h4/list/MAP f) l) g = h4/list/LIST__BIND l (happ (happ h4/combin/o g) f)
% Assm [h4s_lists_LISTu_u_BINDu_u_THMu_c1]: !t h f. h4/list/LIST__BIND (happ (happ h4/list/CONS h) t) f = h4/list/APPEND (happ f h) (h4/list/LIST__BIND t f)
% Assm [h4s_lists_MAPu_u_LISTu_u_BIND]: !l g f. happ (happ h4/list/MAP f) (h4/list/LIST__BIND l g) = h4/list/LIST__BIND l (happ (happ h4/combin/o (happ h4/list/MAP f)) g)
% Assm [h4s_lists_LISTu_u_BINDu_u_THMu_c0]: !f. h4/list/LIST__BIND h4/list/NIL f = h4/list/NIL
% Assm [h4s_lists_APPENDu_u_11u_c1]: !l3 l2 l1. h4/list/APPEND l2 l1 = h4/list/APPEND l3 l1 <=> l2 = l3
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_listu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 r) <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r)
% Assm [h4s_lists_MAP0u_c1]: !t h f. happ (happ h4/list/MAP f) (happ (happ h4/list/CONS h) t) = happ (happ h4/list/CONS (happ f h)) (happ (happ h4/list/MAP f) t)
% Assm [h4s_lists_MAP0u_c0]: !f. happ (happ h4/list/MAP f) h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (h4/combin/C f x) y = happ (happ f y) x
% Assm [h4s_lists_MAPu_u_MAPu_u_o]: !l g f. happ (happ h4/list/MAP f) (happ (happ h4/list/MAP g) l) = happ (happ h4/list/MAP (happ (happ h4/combin/o f) g)) l
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Goal: !g f. happ (happ h4/combin/o (h4/ring/ring__RN__fupd f)) (h4/ring/ring__R0__fupd g) = happ (happ h4/combin/o (h4/ring/ring__R0__fupd g)) (h4/ring/ring__RN__fupd f)
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_Q1390548,TV_Q1390544]: ![V_f, V_g]: (![V_x]: s(TV_Q1390544,happ(s(t_fun(TV_Q1390548,TV_Q1390544),V_f),s(TV_Q1390548,V_x))) = s(TV_Q1390544,happ(s(t_fun(TV_Q1390548,TV_Q1390544),V_g),s(TV_Q1390548,V_x))) => s(t_fun(TV_Q1390548,TV_Q1390544),V_f) = s(t_fun(TV_Q1390548,TV_Q1390544),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_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typering0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_x, V_xi_, V_x0, V_xi_0, V_x1]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering0),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0))),s(t_fun(TV_u_27a,TV_u_27a),V_x1))) = s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_xi_),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0),s(t_fun(TV_u_27a,TV_u_27a),V_x1))))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))))))).
fof(ah4s_rings_ringu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3, V_a4]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3),s(t_fun(TV_u_27a,TV_u_27a),V_a4))))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering2(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_r)))).
fof(ah4s_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_rings_ring1, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering0)).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_rings_ringu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_rings_ring(TV_u_27a),h4s_rings_u_20u_40indu_u_typering1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_rings_u_20u_40indu_u_typering2(s(t_h4s_rings_ring(TV_u_27a),V_a))))) = s(t_h4s_rings_ring(TV_u_27a),V_a)).
fof(ah4s_rings_ringu_u_fupdfupdsu_u_compu_c8, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_g)))))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c4, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f2),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))))).
fof(ah4s_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),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_bools_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_rings_ringu_u_fnu_u_updatesu_c0, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f2),s(TV_u_27a,V_a))))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_RNu_u_fupd0, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f2),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c0, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c6, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_rings_ringu_u_fupdfupdsu_u_compu_c0, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27a),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_rings_ringu_u_fupdcanonu_u_compu_c2, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_c6, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_R0u_u_fupd0, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f2),s(TV_u_27a,V_a))))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c1, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f2),s(TV_u_27a,V_a0))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c3, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_fnu_u_updatesu_c2, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c4, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_g))))),s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c7, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c11, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c9, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c3, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c24, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c7, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c15, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c8, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_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_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_rings_ringu_u_fupdfupdsu_u_compu_c9, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_g))))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h)))).
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),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27a),t_fun(TV_u_27d,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27a),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),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27d,TV_u_27c),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),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),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_rings_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_rings_ringu_u_accessorsu_c0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a)).
fof(ah4s_rings_ringu_u_accessorsu_c4, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,TV_u_27a),V_f1)).
fof(ah4s_rings_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a0)).
fof(ah4s_rings_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_rings_ringu_u_fupdfupdsu_c0, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c7, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h)))))).
fof(ah4s_rings_ringu_u_fupdfupdsu_u_compu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27a),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_rings_ring(TV_u_27a)),V_h)))).
fof(ah4s_rings_ringu_u_accfupdsu_c16, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c8, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c4, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c3, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c20, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c12, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c1, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_c0, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_RM0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_rings_ringu_u_R1u_u_fupd0, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f2),s(TV_u_27a,V_a0))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_R10, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a0)).
fof(ah4s_rings_ringu_u_RMu_u_fupd0, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_RP0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_rings_ringu_u_RPu_u_fupd0, axiom, ![TV_u_27a]: ![V_f2, V_f1, V_f0, V_f, V_a0, V_a]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f2))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1)))).
fof(ah4s_rings_ringu_u_R00, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a)).
fof(ah4s_rings_ringu_u_RN0, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,TV_u_27a),V_f1)).
fof(ah4s_rings_ringu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))) <=> ![V_uu_27ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3, V_a4]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3),s(t_fun(TV_u_27a,TV_u_27a),V_a4))))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_uu_1))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))))))),V_rep))))))).
fof(ah4s_rings_ringu_u_fupdfupdsu_u_compu_c2, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27a),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_rings_ringu_u_fupdcanonu_u_compu_c4, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c1, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a))),h4s_combins_o),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c8, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c5, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c9, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h)))))).
fof(ah4s_rings_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_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27a),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_rings_ring(TV_u_27a)),V_h)))).
fof(ah4s_rings_ringu_u_fupdfupdsu_u_compu_c4, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(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))),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)))),happ(s(t_fun(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))),t_fun(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))),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_rings_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_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(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))),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)))),happ(s(t_fun(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))),t_fun(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))),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_rings_ring(TV_u_27a)),V_h)))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c3, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(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))),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)))),happ(s(t_fun(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))),t_fun(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))),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdfupdsu_c2, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(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))),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)))),happ(s(t_fun(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))),t_fun(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))),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c10, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_rings_ringu_u_fupdcanonu_u_compu_c11, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h)))))).
fof(ah4s_rings_ringu_u_fupdfupdsu_u_compu_c5, axiom, ![TV_u_27b,TV_u_27a]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))),s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),V_h))))) = s(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)),t_fun(TV_u_27b,t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(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))),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)))),happ(s(t_fun(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))),t_fun(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))),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_rings_ring(TV_u_27a)),V_h)))).
fof(ah4s_rings_ringu_u_fupdfupdsu_u_compu_c6, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),happ(s(t_fun(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))),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)))),happ(s(t_fun(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))),t_fun(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))),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_rings_ringu_u_sizeu_u_def, axiom, ![TV_u_27a]: ![V_f, V_a4, V_a3, V_a2, V_a1, V_a0]: s(t_h4s_nums_num,h4s_rings_ringu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,TV_u_27a),t_h4s_rings_ring(TV_u_27a)))))),h4s_rings_ring0),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))),s(t_fun(TV_u_27a,TV_u_27a),V_a4))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a1)))))))).
fof(ah4s_rings_ringu_u_accfupdsu_c13, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c21, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_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_rings_ringu_u_accfupdsu_c9, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_accfupdsu_c0, axiom, ![TV_u_27a]: ![V_r, V_f]: s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c2, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rpu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accfupdsu_c17, axiom, ![TV_u_27a]: ![V_r, V_f]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r)))).
fof(ah4s_rings_ringu_u_fupdcanonu_c4, axiom, ![TV_u_27a]: ![V_r, V_g, V_f]: s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(t_h4s_rings_ring(TV_u_27a),happ(s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_lists_LISTu_u_APPLYu_u_o, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_xs, V_gs, V_fs]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27a)),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a))),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))),t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a))))),happ(s(t_fun(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a))),t_fun(t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))),t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))))),h4s_lists_cons),s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a))),h4s_combins_o))),s(t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))),h4s_lists_nil))),s(t_h4s_lists_list(t_fun(TV_u_27c,TV_u_27a)),V_fs))),s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27c)),V_gs))),s(t_h4s_lists_list(TV_u_27b),V_xs))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(TV_u_27c,TV_u_27a)),V_fs),s(t_h4s_lists_list(TV_u_27c),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27c)),V_gs),s(t_h4s_lists_list(TV_u_27b),V_xs)))))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_o, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27c)),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)))),h4s_options_some(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o))),s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27b)),V_g))),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27b)),V_g),s(t_h4s_options_option(TV_u_27a),V_x)))))).
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),happ(s(t_fun(t_fun(TV_u_27z,TV_u_27z),t_fun(TV_u_27z,TV_u_27z)),happ(s(t_fun(t_fun(TV_u_27z,TV_u_27z),t_fun(t_fun(TV_u_27z,TV_u_27z),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_semiu_u_rings_semiu_u_ringu_u_fupdcanonu_u_compu_c8, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),t_fun(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f))))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),t_fun(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_sr1u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srmu_u_fupd(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),V_f)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (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_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_some(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(t_h4s_options_option(TV_u_27b),V_x)))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_APPLYu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_none),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
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_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27semiu_u_ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r)))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_a))))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_a)).
fof(ah4s_semiu_u_rings_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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_lists_LISTu_u_APPLYu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_xs, V_fs]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27a)),V_fs),s(t_h4s_lists_list(TV_u_27b),V_xs))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27a)),V_fs),s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_combins_c(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),h4s_lists_map),s(t_h4s_lists_list(TV_u_27b),V_xs)))))).
fof(ah4s_lists_SINGLu_u_LISTu_u_APPLYu_u_L, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_x))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f),s(TV_u_27b,V_x)))).
fof(ah4s_lists_LISTu_u_BINDu_u_APPEND, axiom, ![TV_u_27b,TV_u_27a]: ![V_l2, V_l1, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),V_l1),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),V_l2),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f)))))).
fof(ah4s_lists_LISTu_u_BINDu_u_MAP, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_l, V_g, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27b))),h4s_lists_map),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27c),V_l))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_g))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27c),V_l),s(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a)))),h4s_combins_o),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_f)))))).
fof(ah4s_lists_LISTu_u_BINDu_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),V_t),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f)))))).
fof(ah4s_lists_MAPu_u_LISTu_u_BIND, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_l, V_g, V_f]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),h4s_lists_map),s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27c),V_l),s(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27b)),V_g))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27c),V_l),s(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27b)),t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),t_fun(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27b)),t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),h4s_lists_map),s(t_fun(TV_u_27b,TV_u_27a),V_f))))),s(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27b)),V_g)))))).
fof(ah4s_lists_LISTu_u_BINDu_u_THMu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_APPENDu_u_11u_c1, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l2),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l3),s(t_h4s_lists_list(TV_u_27a),V_l1))) <=> s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),V_l3))).
fof(ah4s_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
fof(ah4s_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_lists_listu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27listu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_lists_map),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_lists_map),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_lists_map),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27b,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_y))),s(TV_u_27b,V_x)))).
fof(ah4s_lists_MAPu_u_MAPu_u_o, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_l, V_g, V_f]: s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c))),h4s_lists_map),s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_lists_map),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27c))),h4s_lists_map),s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o),s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g))))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ch4s_rings_ringu_u_fupdcanonu_u_compu_c12, conjecture, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))) = s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_r0u_u_fupd(s(t_fun(TV_u_27a,TV_u_27a),V_g))))),s(t_fun(t_h4s_rings_ring(TV_u_27a),t_h4s_rings_ring(TV_u_27a)),h4s_rings_ringu_u_rnu_u_fupd(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,TV_u_27a)),V_f)))))).
