%   ORIGINAL: h4/semi__ring/semi__ring__accessors_c3
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/semi__ring/semi__ring__SRM0: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/semi__ring/semi__ring1: h4/semi__ring/semi__ring0 = h4/semi__ring/_20_40ind__typesemi__ring0
% Assm: h4/semi__ring/semi__ring__SRP0: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm: h4/semi__ring/semi__ring__SR00: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/semi__ring/semi__ring__SR10: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring0 a a0 f f0) = a0
% Assm: h4/semi__ring/semi__ring__repfns_c1: !r. (\a0_27. !_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3) ==> _27semi__ring_27 a0_270) ==> _27semi__ring_27 a0_27) r <=> h4/semi__ring/_20_40ind__typesemi__ring2 (h4/semi__ring/_20_40ind__typesemi__ring1 r) = r
% Assm: h4/semi__ring/hidden____20__40ind____typesemi____ring0____def: h4/semi__ring/_20_40ind__typesemi__ring0 = (\a0 a1 a2 a3. h4/semi__ring/_20_40ind__typesemi__ring1 ((\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3))
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% 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/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/semi__ring/semi__ring__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 (h4/pair/_2C a10 (h4/pair/_2C a20 a30))) (\n. h4/ind__type/BOTTOM)) a0 a1 a2 a3) ==> _27semi__ring_27 a0_270) ==> _27semi__ring_27 a0_27) rep
% Assm: h4/semi__ring/semi__ring__accessors_c0: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/semi__ring/semi__ring__accessors_c2: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm: h4/semi__ring/semi__ring__size__def: !f a3 a2 a1 a0. h4/semi__ring/semi__ring__size f (h4/semi__ring/semi__ring0 a0 a1 a2 a3) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (f a1))
% Assm: h4/semi__ring/semi__ring__accessors_c1: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (h4/semi__ring/semi__ring0 a a0 f f0) = a0
% Assm: h4/bool/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/operator/ASSOC__DEF: !f. h4/operator/ASSOC f <=> (!x y z. f x (f y z) = f (f x y) z)
% Assm: h4/semi__ring/semi__ring__case__def: !f a3 a2 a1 a0. h4/semi__ring/semi__ring__CASE (h4/semi__ring/semi__ring0 a0 a1 a2 a3) f = f a0 a1 a2 a3
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/operator/ASSOC__SYM: !f. h4/operator/ASSOC f <=> (!x y z. f (f x y) z = f x (f y z))
% Assm: h4/operator/MONOID__DEF: !f e. h4/operator/MONOID f e <=> h4/operator/ASSOC f /\ h4/operator/RIGHT__ID f e /\ h4/operator/LEFT__ID f e
% Assm: h4/list/FOLDL__EQ__FOLDR: !l f e. h4/operator/ASSOC f /\ h4/operator/COMM f ==> h4/list/FOLDL f e l = h4/list/FOLDR f e l
% Assm: h4/operator/FCOMM__ASSOC: !f. h4/operator/FCOMM f f <=> h4/operator/ASSOC f
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/operator/FCOMM__DEF: !g f. h4/operator/FCOMM f g <=> (!x y z. g x (f y z) = f (g x y) z)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm: h4/list/FOLDR0_c0: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm: h4/list/FOLDR0_c1: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = f x (h4/list/FOLDR f e l)
% Assm: h4/operator/COMM__DEF: !f. h4/operator/COMM f <=> (!x y. f x y = f y x)
% Assm: h4/bool/RIGHT__FORALL__IMP__THM: !Q P. (!x. P ==> Q x) <=> P ==> (!x. Q x)
% Assm: h4/list/FOLDL0_c1: !x l f e. h4/list/FOLDL f e (h4/list/CONS x l) = h4/list/FOLDL f (f e x) l
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/MEM__APPEND: !l2 l1 e. h4/bool/IN e (h4/list/LIST__TO__SET (h4/list/APPEND l1 l2)) <=> h4/bool/IN e (h4/list/LIST__TO__SET l1) \/ h4/bool/IN e (h4/list/LIST__TO__SET l2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% 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/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% 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/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/list/LIST__TO__SET0_c0: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/list/LIST__TO__SET0_c1: !t h. h4/list/LIST__TO__SET (h4/list/CONS h t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/list/mem__exists__set: !y x l. h4/bool/IN (h4/pair/_2C x y) (h4/list/LIST__TO__SET l) ==> (?z. x = h4/pair/FST z /\ h4/bool/IN z (h4/list/LIST__TO__SET l))
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/list/MEM_c0: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm: h4/list/MEM_c1: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/list/LAST__APPEND__CONS: !l2 l1 h. h4/list/LAST (h4/list/APPEND l1 (h4/list/CONS h l2)) = h4/list/LAST (h4/list/CONS h l2)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/list/LAST__CONS__cond: !t h. h4/list/LAST (h4/list/CONS h t) = h4/bool/COND (t = h4/list/NIL) h (h4/list/LAST t)
% Assm: h4/list/APPEND__eq__NIL_c0: !l2 l1. h4/list/NIL = h4/list/APPEND l1 l2 <=> l1 = h4/list/NIL /\ l2 = h4/list/NIL
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/relation/TC__CASES2: !z x R. h4/relation/TC R x z <=> R x z \/ (?y. h4/relation/TC R x y /\ R y z)
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/relation/TC__CASES2__E: !z x R. h4/relation/TC R x z ==> R x z \/ (?y. h4/relation/TC R x y /\ R y z)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/TC__RULES_c1: !z y x R. h4/relation/TC R x y /\ h4/relation/TC R y z ==> h4/relation/TC R x z
% Assm: h4/relation/TC__RULES_c0: !y x R. R x y ==> h4/relation/TC R x y
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/relation/WF__EQ__INDUCTION__THM: !R. h4/relation/WF R <=> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/relation/INDUCTION__WF__THM: !R. (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x)) ==> h4/relation/WF R
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/list/FRONT__CONS__EQ__NIL_c1: !xs x. h4/list/NIL = h4/list/FRONT (h4/list/CONS x xs) <=> xs = h4/list/NIL
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/list/FRONT__CONS_c1: !z y x. h4/list/FRONT (h4/list/CONS x (h4/list/CONS y z)) = h4/list/CONS x (h4/list/FRONT (h4/list/CONS y z))
% Assm: h4/list/FRONT__CONS_c0: !x. h4/list/FRONT (h4/list/CONS x h4/list/NIL) = h4/list/NIL
% Assm: h4/list/NOT__NIL__CONS: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/list/NULL__DEF_c1: !t h. h4/list/NULL (h4/list/CONS h t) <=> F
% Assm: h4/list/NULL__DEF_c0: h4/list/NULL h4/list/NIL <=> T
% Assm: h4/while/OWHILE__INV__IND: !s f P G. P s /\ (!x. P x /\ G x ==> P (f x)) ==> (!s_27. h4/while/OWHILE G f s = h4/option/SOME s_27 ==> P s_27)
% Assm: h4/while/OWHILE__def: !s f G. h4/while/OWHILE G f s = h4/bool/COND (?n. ~G (h4/arithmetic/FUNPOW f n s)) (h4/option/SOME (h4/arithmetic/FUNPOW f (h4/while/LEAST (\n. ~G (h4/arithmetic/FUNPOW f n s))) s)) h4/option/NONE
% Assm: h4/while/LEAST__ELIM: !Q P. (?n. P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~P m) /\ P n ==> Q n) ==> Q (h4/while/LEAST P)
% Assm: h4/arithmetic/FUNPOW0_c0: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/arithmetic/FUNPOW0_c1: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (f x)
% Goal: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SRM0]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f0
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_semiu_u_rings_semiu_u_ring1]: h4/semi__ring/semi__ring0 = h4/semi__ring/_20_40ind__typesemi__ring0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SRP0]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SR00]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_SR10]: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a0
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 a3))) _0) ==> happ _27semi__ring_27 a0_270) ==> happ _27semi__ring_27 r) <=> h4/semi__ring/_20_40ind__typesemi__ring2 (h4/semi__ring/_20_40ind__typesemi__ring1 r) = r)
% Assm [h4s_semiu_u_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesemiu_u_u_u_ring0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x' x x'. happ (happ (happ (happ h4/semi__ring/_20_40ind__typesemi__ring0 x) x') x) x' = h4/semi__ring/_20_40ind__typesemi__ring1 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x (h4/pair/_2C x' (h4/pair/_2C x x'))) _0))
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% 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_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27semi__ring_27. (!a0_270. (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 (h4/pair/_2C a1 (h4/pair/_2C a2 a3))) _1) ==> happ _27semi__ring_27 a0_270) ==> happ _27semi__ring_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_sizeu_u_def]: !f a3 a2 a1 a0. h4/semi__ring/semi__ring__size f (happ (happ (happ (happ h4/semi__ring/semi__ring0 a0) a1) a2) a3) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (happ f a1))
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c1]: !f0 f a0 a. h4/semi__ring/semi__ring__SR1 (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = a0
% Assm [h4s_bools_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_operators_ASSOCu_u_DEF]: !f. h4/operator/ASSOC f <=> (!x y z. happ (happ f x) (happ (happ f y) z) = happ (happ f (happ (happ f x) y)) z)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_caseu_u_def]: !f a3 a2 a1 a0. h4/semi__ring/semi__ring__CASE (happ (happ (happ (happ h4/semi__ring/semi__ring0 a0) a1) a2) a3) f = happ (happ (happ (happ f a0) a1) a2) a3
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_operators_ASSOCu_u_SYM]: !f. h4/operator/ASSOC f <=> (!x y z. happ (happ f (happ (happ f x) y)) z = happ (happ f x) (happ (happ f y) z))
% Assm [h4s_operators_MONOIDu_u_DEF]: !f e. h4/operator/MONOID f e <=> h4/operator/ASSOC f /\ h4/operator/RIGHT__ID f e /\ h4/operator/LEFT__ID f e
% Assm [h4s_lists_FOLDLu_u_EQu_u_FOLDR]: !l f e. h4/operator/ASSOC f /\ h4/operator/COMM f ==> h4/list/FOLDL f e l = h4/list/FOLDR f e l
% Assm [h4s_operators_FCOMMu_u_ASSOC]: !f. h4/operator/FCOMM f f <=> h4/operator/ASSOC f
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_operators_FCOMMu_u_DEF]: !g f. h4/operator/FCOMM f g <=> (!x y z. happ (happ g x) (happ (happ f y) z) = happ (happ f (happ (happ g x) y)) z)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_lists_FOLDL0u_c0]: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm [h4s_lists_FOLDR0u_c0]: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm [h4s_lists_FOLDR0u_c1]: !x l f e. h4/list/FOLDR f e (happ (happ h4/list/CONS x) l) = happ (happ f x) (h4/list/FOLDR f e l)
% Assm [h4s_operators_COMMu_u_DEF]: !f. h4/operator/COMM f <=> (!x y. happ (happ f x) y = happ (happ f y) x)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. P ==> happ Q x) <=> P ==> (!x. happ Q x)
% Assm [h4s_lists_FOLDL0u_c1]: !x l f e. h4/list/FOLDL f e (happ (happ h4/list/CONS x) l) = h4/list/FOLDL f (happ (happ f e) x) l
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (happ (happ h4/list/CONS h) t))) ==> (!l. happ P l)
% Assm [h4s_lists_MEMu_u_APPEND]: !l2 l1 e. h4/bool/IN e (h4/list/LIST__TO__SET (h4/list/APPEND l1 l2)) <=> h4/bool/IN e (h4/list/LIST__TO__SET l1) \/ h4/bool/IN e (h4/list/LIST__TO__SET l2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% 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_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% 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_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% 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_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (happ (happ h4/list/CONS h) l1) l2 = happ (happ h4/list/CONS h) (h4/list/APPEND l1 l2)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c0]: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c1]: !t h. h4/list/LIST__TO__SET (happ (happ h4/list/CONS h) t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_lists_memu_u_existsu_u_set]: !y x l. h4/bool/IN (h4/pair/_2C x y) (h4/list/LIST__TO__SET l) ==> (?z. x = h4/pair/FST z /\ h4/bool/IN z (h4/list/LIST__TO__SET l))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_lists_MEMu_c0]: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm [h4s_lists_MEMu_c1]: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (happ (happ h4/list/CONS h) t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_lists_LASTu_u_APPENDu_u_CONS]: !l2 l1 h. h4/list/LAST (h4/list/APPEND l1 (happ (happ h4/list/CONS h) l2)) = h4/list/LAST (happ (happ h4/list/CONS h) l2)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_lists_LASTu_u_CONSu_u_cond]: !t h. ?v. (v <=> t = h4/list/NIL) /\ h4/list/LAST (happ (happ h4/list/CONS h) t) = h4/bool/COND v h (h4/list/LAST t)
% Assm [h4s_lists_APPENDu_u_equ_u_NILu_c0]: !l2 l1. h4/list/NIL = h4/list/APPEND l1 l2 <=> l1 = h4/list/NIL /\ l2 = h4/list/NIL
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_relations_TCu_u_CASES2]: !z x R. h4/relation/TC R x z <=> happ (happ R x) z \/ (?y. h4/relation/TC R x y /\ happ (happ R y) z)
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_relations_TCu_u_CASES2u_u_E]: !z x R. h4/relation/TC R x z ==> happ (happ R x) z \/ (?y. h4/relation/TC R x y /\ happ (happ R y) z)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_TCu_u_RULESu_c1]: !z y x R. h4/relation/TC R x y /\ h4/relation/TC R y z ==> h4/relation/TC R x z
% Assm [h4s_relations_TCu_u_RULESu_c0]: !y x R. happ (happ R x) y ==> h4/relation/TC R x y
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_relations_WFu_u_EQu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R <=> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_relations_INDUCTIONu_u_WFu_u_THM]: !R. (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x)) ==> h4/relation/WF R
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_lists_FRONTu_u_CONSu_u_EQu_u_NILu_c1]: !xs x. h4/list/NIL = h4/list/FRONT (happ (happ h4/list/CONS x) xs) <=> xs = h4/list/NIL
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_lists_FRONTu_u_CONSu_c1]: !z y x. h4/list/FRONT (happ (happ h4/list/CONS x) (happ (happ h4/list/CONS y) z)) = happ (happ h4/list/CONS x) (h4/list/FRONT (happ (happ h4/list/CONS y) z))
% Assm [h4s_lists_FRONTu_u_CONSu_c0]: !x. h4/list/FRONT (happ (happ h4/list/CONS x) h4/list/NIL) = h4/list/NIL
% Assm [h4s_lists_NOTu_u_NILu_u_CONS]: !a1 a0. ~(h4/list/NIL = happ (happ h4/list/CONS a0) a1)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_lists_NULLu_u_DEFu_c1]: !t h. h4/list/NULL (happ (happ h4/list/CONS h) t) <=> F
% Assm [h4s_lists_NULLu_u_DEFu_c0]: h4/list/NULL h4/list/NIL <=> T
% Assm [h4s_whiles_OWHILEu_u_INVu_u_IND]: !s f P G. happ P s /\ (!x. happ P x /\ happ G x ==> happ P (happ f x)) ==> (!s_27. h4/while/OWHILE G f s = h4/option/SOME s_27 ==> happ P s_27)
% Assm [h4s_whiles_OWHILEu_u_def]: !_0. (!G f s n. happ (happ (happ (happ _0 G) f) s) n <=> ~happ G (h4/arithmetic/FUNPOW f n s)) ==> (!s f G. ?v. (v <=> (?n. ~happ G (h4/arithmetic/FUNPOW f n s))) /\ h4/while/OWHILE G f s = h4/bool/COND v (h4/option/SOME (h4/arithmetic/FUNPOW f (h4/while/LEAST (happ (happ (happ _0 G) f) s)) s)) h4/option/NONE)
% Assm [h4s_whiles_LEASTu_u_ELIM]: !Q P. (?n. happ P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~happ P m) /\ happ P n ==> happ Q n) ==> happ Q (h4/while/LEAST P)
% Assm [h4s_arithmetics_FUNPOW0u_c0]: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_arithmetics_FUNPOW0u_c1]: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (happ f x)
% Goal: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (happ (happ (happ (happ h4/semi__ring/semi__ring0 a) a0) f) f0) = f0
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f1)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f1))).
fof(aHLu_EXT, axiom, ![TV_Q1088178,TV_Q1088174]: ![V_f, V_g]: (![V_x]: s(TV_Q1088174,happ(s(t_fun(TV_Q1088178,TV_Q1088174),V_f),s(TV_Q1088178,V_x))) = s(TV_Q1088174,happ(s(t_fun(TV_Q1088178,TV_Q1088174),V_g),s(TV_Q1088178,V_x))) => s(t_fun(TV_Q1088178,TV_Q1088174),V_f) = s(t_fun(TV_Q1088178,TV_Q1088174),V_g))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SRM0, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_semiu_u_rings_semiu_u_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_semiu_u_rings_semiu_u_ringu_u_SRP0, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SR00, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_SR10, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a0)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27semiu_u_ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring2(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_r)))).
fof(ah4s_semiu_u_rings_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesemiu_u_u_u_ring0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_x, V_xi_, V_x0, V_xi_0]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring0),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_u_20u_40indu_u_typesemiu_u_ring1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_xi_),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_x0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_xi_0))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_0))))))).
fof(ah4s_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_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_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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))) <=> ![V_uu_27semiu_u_ringu_27]: (![V_a0u_270]: (?[V_a0, V_a1, V_a2, V_a3]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_uu_1))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_27semiu_u_ringu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)))))),t_bool),V_uu_0),s(t_fun(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a))))))),V_rep))))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c0, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_sizeu_u_def, axiom, ![TV_u_27a]: ![V_f, V_a3, V_a2, V_a1, V_a0]: s(t_h4s_nums_num,h4s_semiu_u_rings_semiu_u_ringu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a1)))))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(TV_u_27a,V_a0)).
fof(ah4s_bools_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_operators_ASSOCu_u_DEF, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) <=> ![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))),s(TV_u_27a,V_z))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_caseu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_a3, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_semiu_u_rings_semiu_u_ringu_u_case(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b)))),V_f))) = s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),TV_u_27b)))),V_f),s(TV_u_27a,V_a0))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_a3)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_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_operators_ASSOCu_u_SYM, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) <=> ![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))),s(TV_u_27a,V_z))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))))).
fof(ah4s_operators_MONOIDu_u_DEF, axiom, ![TV_u_27a]: ![V_f, V_e]: (p(s(t_bool,h4s_operators_monoid(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))) <=> (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) & (p(s(t_bool,h4s_operators_rightu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))) & p(s(t_bool,h4s_operators_leftu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))))))).
fof(ah4s_lists_FOLDLu_u_EQu_u_FOLDR, axiom, ![TV_u_27a]: ![V_l, V_f, V_e]: ((p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) & p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))) => s(TV_u_27a,h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_operators_FCOMMu_u_ASSOC, axiom, ![TV_u_27a]: ![V_f]: s(t_bool,h4s_operators_fcomm(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_f))) = s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_operators_FCOMMu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_g, V_f]: (p(s(t_bool,h4s_operators_fcomm(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_fun(TV_u_27c,t_fun(TV_u_27a,TV_u_27a)),V_g)))) <=> ![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27c,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27b,V_z))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))),s(TV_u_27b,V_z))))).
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_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_lists_FOLDL0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_lists_FOLDR0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_lists_FOLDR0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_operators_COMMu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f)))) <=> ![V_x, V_y]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_lists_FOLDL0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e))),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_lists_listu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_MEMu_u_APPEND, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1)))))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l2))))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f1)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_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_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_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_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
fof(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_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_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(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_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f1)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(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_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_h),s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),V_t)))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_lists_memu_u_existsu_u_set, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_l]: (p(s(t_bool,h4s_bools_in(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(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l)))))) => ?[V_z]: (s(TV_u_27a,V_x) = s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_z))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_z),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f1))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f1))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f1))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f1))) => p(s(t_bool,f1))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f1))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f1)))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f1))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f1))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f1)))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f1)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_lists_MEMu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))) = s(t_bool,f1)).
fof(ah4s_lists_MEMu_c1, axiom, ![TV_u_27a]: ![V_x, V_t, V_h]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_h) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_t))))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_lists_LASTu_u_APPENDu_u_CONS, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),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_l2))))))) = s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f1))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f1))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f1)))).
fof(ah4s_lists_LASTu_u_CONSu_u_cond, axiom, ![TV_u_27a]: ![V_t, V_h]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27a),V_t) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)) & s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,V_h),s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_APPENDu_u_equ_u_NILu_c0, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(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_l1) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) & s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f1) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, 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_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f1))) <=> p(s(t_bool,f1)))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f1),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_relations_TCu_u_CASES2, axiom, ![TV_u_27a]: ![V_z, V_x, V_R]: (p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_z)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))) | ?[V_y]: (p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,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_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f1))))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) & ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_TCu_u_CASES2u_u_E, axiom, ![TV_u_27a]: ![V_z, V_x, V_R]: (p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_z)))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))) | ?[V_y]: (p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_TCu_u_RULESu_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_R]: ((p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) & p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y),s(TV_u_27a,V_z))))) => p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_z)))))).
fof(ah4s_relations_TCu_u_RULESu_c0, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_relations_WFu_u_EQu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_INDUCTIONu_u_WFu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_lists_FRONTu_u_CONSu_u_EQu_u_NILu_c1, axiom, ![TV_u_27a]: ![V_xs, V_x]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_front(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_x))),s(t_h4s_lists_list(TV_u_27a),V_xs))))) <=> s(t_h4s_lists_list(TV_u_27a),V_xs) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_FRONTu_u_CONSu_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_front(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_x))),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_y))),s(t_h4s_lists_list(TV_u_27a),V_z))))))) = 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_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_front(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_y))),s(t_h4s_lists_list(TV_u_27a),V_z)))))))).
fof(ah4s_lists_FRONTu_u_CONSu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_front(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_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_NOTu_u_NILu_u_CONS, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = 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_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f1) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_NULLu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_bool,h4s_lists_null(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_bool,f1)).
fof(ah4s_lists_NULLu_u_DEFu_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_whiles_OWHILEu_u_INVu_u_IND, axiom, ![TV_u_27a]: ![V_s, V_f, V_P, V_G]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s)))) & ![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_G),s(TV_u_27a,V_x))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))))))) => ![V_su_27]: (s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_su_27))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_su_27))))))).
fof(ah4s_whiles_OWHILEu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_G, V_f, V_s, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_G))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(TV_u_27a,V_s))),s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_s)))))))) => ![V_s, V_f, V_G]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_n]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_s)))))))) & s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_G))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(TV_u_27a,V_s))))),s(TV_u_27a,V_s))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))).
fof(ah4s_whiles_LEASTu_u_ELIM, axiom, ![V_Q, V_P]: ((?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_n]: ((![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,V_n)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_arithmetics_FUNPOW0u_c0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_FUNPOW0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))))).
fof(ch4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3, conjecture, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a))))),h4s_semiu_u_rings_semiu_u_ring0),s(TV_u_27a,V_a))),s(TV_u_27a,V_a0))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
