%   ORIGINAL: h4/rich__list/MAP__FOLDL
% 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/TRUTH: T
% Assm: h4/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% 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/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/list/FOLDL__SNOC: !x l f e. h4/list/FOLDL f e (h4/list/SNOC x l) = f (h4/list/FOLDL f e l) x
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% 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/rich__list/SUFFIX__DEF: !l P. h4/rich__list/SUFFIX P l = h4/list/FOLDL (\l_27 x. h4/bool/COND (P x) (h4/list/SNOC x l_27) h4/list/NIL) h4/list/NIL l
% Assm: h4/list/MAP__SNOC: !x l f. h4/list/MAP f (h4/list/SNOC x l) = h4/list/SNOC (f x) (h4/list/MAP f l)
% Assm: h4/list/SUM__MAP__FOLDL: !ls f. h4/list/SUM (h4/list/MAP f ls) = h4/list/FOLDL (\a e. h4/arithmetic/_2B a (f e)) h4/num/0 ls
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/FOLDL__UNION__BIGUNION: !s ls f. h4/list/FOLDL (\s0 x. h4/pred__set/UNION s0 (f x)) s ls = h4/pred__set/UNION s (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f (h4/list/LIST__TO__SET ls)))
% Assm: h4/list/FOLDL__CONG: !l_27 l f_27 f b_27 b. l = l_27 /\ b = b_27 /\ (!x a. h4/bool/IN x (h4/list/LIST__TO__SET l_27) ==> f a x = f_27 a x) ==> h4/list/FOLDL f b l = h4/list/FOLDL f_27 b_27 l_27
% Assm: h4/list/FOLDL__ZIP__SAME: !ls f e. h4/list/FOLDL f e (h4/list/ZIP (h4/pair/_2C ls ls)) = h4/list/FOLDL (\x y. f x (h4/pair/_2C y y)) e ls
% Assm: h4/list/FOLDL2__FOLDL: !l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> (!f a. h4/list/FOLDL2 f a l1 l2 = h4/list/FOLDL (\a0. h4/pair/UNCURRY (f a0)) a (h4/list/ZIP (h4/pair/_2C l1 l2)))
% Assm: h4/list/FOLDL__UNION__BIGUNION__paired: !s ls f. h4/list/FOLDL (\s0. h4/pair/UNCURRY (\x y. h4/pred__set/UNION s0 (f x y))) s ls = h4/pred__set/UNION s (h4/pred__set/BIGUNION (h4/pred__set/IMAGE (h4/pair/UNCURRY f) (h4/list/LIST__TO__SET ls)))
% 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/rich__list/LENGTH__FOLDL: !l. h4/list/LENGTH l = h4/list/FOLDL (\l_27 x. h4/num/SUC l_27) h4/num/0 l
% Assm: h4/rich__list/FOLDR__FOLDL: !f e. h4/operator/MONOID f e ==> (!l. h4/list/FOLDR f e l = h4/list/FOLDL f e l)
% Assm: h4/list/ITSET__eq__FOLDL__SET__TO__LIST: !s. h4/pred__set/FINITE s ==> (!f a. h4/pred__set/ITSET f s a = h4/list/FOLDL (h4/combin/C f) a (h4/list/SET__TO__LIST s))
% 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/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/rich__list/MAP__FOLDR: !l f. h4/list/MAP f l = h4/list/FOLDR (\x l_27. h4/list/CONS (f x) l_27) h4/list/NIL l
% Assm: h4/list/EXISTS__LIST__EQ__MAP: !ls f. h4/list/EVERY (\x. ?y. x = f y) ls ==> (?l. ls = h4/list/MAP f l)
% Assm: h4/list/LUPDATE__MAP: !x n l f. h4/list/MAP f (h4/list/LUPDATE x n l) = h4/list/LUPDATE (f x) n (h4/list/MAP f l)
% Assm: h4/list/ALL__DISTINCT__MAP__INJ: !ls f. (!x y. h4/bool/IN x (h4/list/LIST__TO__SET ls) /\ h4/bool/IN y (h4/list/LIST__TO__SET ls) /\ f x = f y ==> x = y) /\ h4/list/ALL__DISTINCT ls ==> h4/list/ALL__DISTINCT (h4/list/MAP f ls)
% Assm: h4/list/SNOC__INDUCT: !P. P h4/list/NIL /\ (!l. P l ==> (!x. P (h4/list/SNOC x l))) ==> (!l. P l)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% 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/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/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% 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/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/list/SNOC__APPEND: !x l. h4/list/SNOC x l = h4/list/APPEND l (h4/list/CONS x h4/list/NIL)
% Assm: h4/list/MAP__APPEND: !l2 l1 f. h4/list/MAP f (h4/list/APPEND l1 l2) = h4/list/APPEND (h4/list/MAP f l1) (h4/list/MAP f l2)
% Assm: h4/list/ZIP0_c1: !x2 x1 l2 l1. h4/list/ZIP (h4/pair/_2C (h4/list/CONS x1 l1) (h4/list/CONS x2 l2)) = h4/list/CONS (h4/pair/_2C x1 x2) (h4/list/ZIP (h4/pair/_2C l1 l2))
% Assm: h4/list/ZIP0_c0: h4/list/ZIP (h4/pair/_2C h4/list/NIL h4/list/NIL) = h4/list/NIL
% Assm: h4/list/SNOC0_c1: !x_27 x l. h4/list/SNOC x (h4/list/CONS x_27 l) = h4/list/CONS x_27 (h4/list/SNOC x l)
% Assm: h4/list/SNOC0_c0: !x. h4/list/SNOC x h4/list/NIL = h4/list/CONS x h4/list/NIL
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/LIST__TO__SET0_c0: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm: h4/pred__set/UNION__ASSOC: !u t s. h4/pred__set/UNION s (h4/pred__set/UNION t u) = h4/pred__set/UNION (h4/pred__set/UNION s t) u
% 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/BIGUNION__EMPTY: h4/pred__set/BIGUNION h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm: h4/pred__set/IMAGE__EMPTY: !f. h4/pred__set/IMAGE f h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm: h4/pred__set/UNION__EMPTY_c1: !s. h4/pred__set/UNION s h4/pred__set/EMPTY = s
% Assm: h4/pred__set/IMAGE__INSERT: !x s f. h4/pred__set/IMAGE f (h4/pred__set/INSERT x s) = h4/pred__set/INSERT (f x) (h4/pred__set/IMAGE f s)
% Assm: h4/pred__set/BIGUNION__INSERT: !s P. h4/pred__set/BIGUNION (h4/pred__set/INSERT s P) = h4/pred__set/UNION s (h4/pred__set/BIGUNION P)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% 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/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/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/list/LENGTH__SNOC: !x l. h4/list/LENGTH (h4/list/SNOC x l) = h4/num/SUC (h4/list/LENGTH l)
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/list/SUM0_c0: h4/list/SUM h4/list/NIL = h4/num/0
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/list/SUM__SNOC: !x l. h4/list/SUM (h4/list/SNOC x l) = h4/arithmetic/_2B (h4/list/SUM l) x
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/operator/LEFT__ID__DEF: !f e. h4/operator/LEFT__ID f e <=> (!x. f e x = x)
% 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/operator/RIGHT__ID__DEF: !f e. h4/operator/RIGHT__ID f e <=> (!x. f x e = x)
% 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/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/list/FOLDL2__def_c0: !f cs c bs b a. h4/list/FOLDL2 f a (h4/list/CONS b bs) (h4/list/CONS c cs) = h4/list/FOLDL2 f (f a b c) bs cs
% Assm: h4/list/FOLDL2__def_c1: !f cs a. h4/list/FOLDL2 f a h4/list/NIL cs = a
% Assm: h4/list/LENGTH__NIL__SYM: !l. h4/num/0 = h4/list/LENGTH l <=> l = h4/list/NIL
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/pred__set/FINITE__COMPLETE__INDUCTION: !P. (!x. (!y. h4/pred__set/PSUBSET y x ==> P y) ==> h4/pred__set/FINITE x ==> P x) ==> (!x. h4/pred__set/FINITE x ==> P x)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/pred__set/ITSET__THM: !s f b. h4/pred__set/FINITE s ==> h4/pred__set/ITSET f s b = h4/bool/COND (s = h4/pred__set/EMPTY) b (h4/pred__set/ITSET f (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b))
% Assm: h4/pred__set/FINITE__EMPTY: h4/pred__set/FINITE h4/pred__set/EMPTY
% Assm: h4/pred__set/REST__PSUBSET: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/PSUBSET (h4/pred__set/REST s) s
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/pred__set/FINITE__REST: !s. h4/pred__set/FINITE s ==> h4/pred__set/FINITE (h4/pred__set/REST s)
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/list/SET__TO__LIST__THM: !s. h4/pred__set/FINITE s ==> h4/list/SET__TO__LIST s = h4/bool/COND (s = h4/pred__set/EMPTY) h4/list/NIL (h4/list/CONS (h4/pred__set/CHOICE s) (h4/list/SET__TO__LIST (h4/pred__set/REST s)))
% 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/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/list/LUPDATE__def_c2: !x n l e. h4/list/LUPDATE e (h4/num/SUC n) (h4/list/CONS x l) = h4/list/CONS x (h4/list/LUPDATE e n l)
% Assm: h4/list/LUPDATE__def_c1: !x l e. h4/list/LUPDATE e h4/num/0 (h4/list/CONS x l) = h4/list/CONS e l
% Assm: h4/list/LUPDATE__def_c0: !n e. h4/list/LUPDATE e n h4/list/NIL = h4/list/NIL
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/list/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/list/MEM_c0: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm: h4/list/ALL__DISTINCT0_c0: h4/list/ALL__DISTINCT h4/list/NIL <=> T
% Assm: h4/list/MEM__MAP: !x l f. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/MAP f l)) <=> (?y. x = f y /\ h4/bool/IN y (h4/list/LIST__TO__SET l))
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/list/ALL__DISTINCT0_c1: !t h. h4/list/ALL__DISTINCT (h4/list/CONS h t) <=> ~h4/bool/IN h (h4/list/LIST__TO__SET t) /\ h4/list/ALL__DISTINCT t
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Goal: !l f. h4/list/MAP f l = h4/list/FOLDL (\l_27 x. h4/list/SNOC (f x) l_27) h4/list/NIL l
%   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_TRUTH]: T
% Assm [h4s_lists_FOLDL0u_c0]: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% 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_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_lists_FOLDLu_u_SNOC]: !x l f e. h4/list/FOLDL f e (h4/list/SNOC x l) = happ (happ f (h4/list/FOLDL f e l)) x
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% 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_richu_u_lists_SUFFIXu_u_DEF]: !_1. (!P l_27 x. happ (happ (happ _1 P) l_27) x = h4/bool/COND (happ P x) (h4/list/SNOC x l_27) h4/list/NIL) ==> (!_0. (!P l_27. happ (happ _0 P) l_27 = happ (happ _1 P) l_27) ==> (!l P. h4/rich__list/SUFFIX P l = h4/list/FOLDL (happ _0 P) h4/list/NIL l))
% Assm [h4s_lists_MAPu_u_SNOC]: !x l f. h4/list/MAP f (h4/list/SNOC x l) = h4/list/SNOC (happ f x) (h4/list/MAP f l)
% Assm [h4s_lists_SUMu_u_MAPu_u_FOLDL]: !_1. (!a f e. happ (happ (happ _1 a) f) e = h4/arithmetic/_2B a (happ f e)) ==> (!_0. (!f a. happ (happ _0 f) a = happ (happ _1 a) f) ==> (!ls f. h4/list/SUM (h4/list/MAP f ls) = h4/list/FOLDL (happ _0 f) h4/num/0 ls))
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_FOLDLu_u_UNIONu_u_BIGUNION]: !_1. (!s0 f x. happ (happ (happ _1 s0) f) x = h4/pred__set/UNION s0 (happ f x)) ==> (!_0. (!f s0. happ (happ _0 f) s0 = happ (happ _1 s0) f) ==> (!s ls f. h4/list/FOLDL (happ _0 f) s ls = h4/pred__set/UNION s (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f (h4/list/LIST__TO__SET ls)))))
% Assm [h4s_lists_FOLDLu_u_CONG]: !l_27 l f_27 f b_27 b. l = l_27 /\ b = b_27 /\ (!x a. h4/bool/IN x (h4/list/LIST__TO__SET l_27) ==> happ (happ f a) x = happ (happ f_27 a) x) ==> h4/list/FOLDL f b l = h4/list/FOLDL f_27 b_27 l_27
% Assm [h4s_lists_FOLDLu_u_ZIPu_u_SAME]: !_1. (!f x y. happ (happ (happ _1 f) x) y = happ (happ f x) (h4/pair/_2C y y)) ==> (!_0. (!f x. happ (happ _0 f) x = happ (happ _1 f) x) ==> (!ls f e. h4/list/FOLDL f e (h4/list/ZIP (h4/pair/_2C ls ls)) = h4/list/FOLDL (happ _0 f) e ls))
% Assm [h4s_lists_FOLDL2u_u_FOLDL]: !_0. (!f a0. happ (happ _0 f) a0 = h4/pair/UNCURRY (happ f a0)) ==> (!l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> (!f a. h4/list/FOLDL2 f a l1 l2 = h4/list/FOLDL (happ _0 f) a (h4/list/ZIP (h4/pair/_2C l1 l2))))
% Assm [h4s_lists_FOLDLu_u_UNIONu_u_BIGUNIONu_u_paired]: !_2. (!s0 f x y. happ (happ (happ (happ _2 s0) f) x) y = h4/pred__set/UNION s0 (happ (happ f x) y)) ==> (!_1. (!s0 f x. happ (happ (happ _1 s0) f) x = happ (happ (happ _2 s0) f) x) ==> (!_0. (!f s0. happ (happ _0 f) s0 = h4/pair/UNCURRY (happ (happ _1 s0) f)) ==> (!s ls f. h4/list/FOLDL (happ _0 f) s ls = h4/pred__set/UNION s (h4/pred__set/BIGUNION (h4/pred__set/IMAGE (h4/pair/UNCURRY f) (h4/list/LIST__TO__SET ls))))))
% 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_richu_u_lists_LENGTHu_u_FOLDL]: !_1. (!l_27 x. happ (happ _1 l_27) x = h4/num/SUC l_27) ==> (!_0. (!l_27. happ _0 l_27 = happ _1 l_27) ==> (!l. h4/list/LENGTH l = h4/list/FOLDL _0 h4/num/0 l))
% Assm [h4s_richu_u_lists_FOLDRu_u_FOLDL]: !f e. h4/operator/MONOID f e ==> (!l. h4/list/FOLDR f e l = h4/list/FOLDL f e l)
% Assm [h4s_lists_ITSETu_u_equ_u_FOLDLu_u_SETu_u_TOu_u_LIST]: !s. h4/pred__set/FINITE s ==> (!f a. h4/pred__set/ITSET f s a = h4/list/FOLDL (h4/combin/C f) a (h4/list/SET__TO__LIST s))
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (happ (happ h4/list/CONS h) t) = happ (happ h4/list/CONS (happ f h)) (h4/list/MAP f t)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_richu_u_lists_MAPu_u_FOLDR]: !_1. (!f x l_27. happ (happ (happ _1 f) x) l_27 = happ (happ h4/list/CONS (happ f x)) l_27) ==> (!_0. (!f x. happ (happ _0 f) x = happ (happ _1 f) x) ==> (!l f. h4/list/MAP f l = h4/list/FOLDR (happ _0 f) h4/list/NIL l))
% Assm [h4s_lists_EXISTSu_u_LISTu_u_EQu_u_MAP]: !_0. (!f x. happ (happ _0 f) x <=> (?y. x = happ f y)) ==> (!ls f. h4/list/EVERY (happ _0 f) ls ==> (?l. ls = h4/list/MAP f l))
% Assm [h4s_lists_LUPDATEu_u_MAP]: !x n l f. h4/list/MAP f (h4/list/LUPDATE x n l) = h4/list/LUPDATE (happ f x) n (h4/list/MAP f l)
% Assm [h4s_lists_ALLu_u_DISTINCTu_u_MAPu_u_INJ]: !ls f. (!x y. h4/bool/IN x (h4/list/LIST__TO__SET ls) /\ h4/bool/IN y (h4/list/LIST__TO__SET ls) /\ happ f x = happ f y ==> x = y) /\ h4/list/ALL__DISTINCT ls ==> h4/list/ALL__DISTINCT (h4/list/MAP f ls)
% Assm [h4s_lists_SNOCu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!l. happ P l ==> (!x. happ P (h4/list/SNOC x l))) ==> (!l. happ P l)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% 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_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_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_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% 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_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% 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_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_lists_SNOCu_u_APPEND]: !x l. h4/list/SNOC x l = h4/list/APPEND l (happ (happ h4/list/CONS x) h4/list/NIL)
% Assm [h4s_lists_MAPu_u_APPEND]: !l2 l1 f. h4/list/MAP f (h4/list/APPEND l1 l2) = h4/list/APPEND (h4/list/MAP f l1) (h4/list/MAP f l2)
% Assm [h4s_lists_ZIP0u_c1]: !x2 x1 l2 l1. h4/list/ZIP (h4/pair/_2C (happ (happ h4/list/CONS x1) l1) (happ (happ h4/list/CONS x2) l2)) = happ (happ h4/list/CONS (h4/pair/_2C x1 x2)) (h4/list/ZIP (h4/pair/_2C l1 l2))
% Assm [h4s_lists_ZIP0u_c0]: h4/list/ZIP (h4/pair/_2C h4/list/NIL h4/list/NIL) = h4/list/NIL
% Assm [h4s_lists_SNOC0u_c1]: !x_27 x l. h4/list/SNOC x (happ (happ h4/list/CONS x_27) l) = happ (happ h4/list/CONS x_27) (h4/list/SNOC x l)
% Assm [h4s_lists_SNOC0u_c0]: !x. h4/list/SNOC x h4/list/NIL = happ (happ h4/list/CONS x) h4/list/NIL
% Assm [h4s_lists_listu_u_induction]: !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_LISTu_u_TOu_u_SET0u_c0]: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_UNIONu_u_ASSOC]: !u t s. h4/pred__set/UNION s (h4/pred__set/UNION t u) = h4/pred__set/UNION (h4/pred__set/UNION s t) u
% 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_BIGUNIONu_u_EMPTY]: h4/pred__set/BIGUNION h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_IMAGEu_u_EMPTY]: !f. h4/pred__set/IMAGE f h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_UNIONu_u_EMPTYu_c1]: !s. h4/pred__set/UNION s h4/pred__set/EMPTY = s
% Assm [h4s_predu_u_sets_IMAGEu_u_INSERT]: !x s f. h4/pred__set/IMAGE f (h4/pred__set/INSERT x s) = h4/pred__set/INSERT (happ f x) (h4/pred__set/IMAGE f s)
% Assm [h4s_predu_u_sets_BIGUNIONu_u_INSERT]: !s P. h4/pred__set/BIGUNION (h4/pred__set/INSERT s P) = h4/pred__set/UNION s (h4/pred__set/BIGUNION P)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% 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_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_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_lists_LENGTHu_u_SNOC]: !x l. h4/list/LENGTH (h4/list/SNOC x l) = h4/num/SUC (h4/list/LENGTH l)
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_lists_SUM0u_c0]: h4/list/SUM h4/list/NIL = h4/num/0
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_lists_SUMu_u_SNOC]: !x l. h4/list/SUM (h4/list/SNOC x l) = h4/arithmetic/_2B (h4/list/SUM l) x
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_operators_LEFTu_u_IDu_u_DEF]: !f e. h4/operator/LEFT__ID f e <=> (!x. happ (happ f e) x = x)
% 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_operators_RIGHTu_u_IDu_u_DEF]: !f e. h4/operator/RIGHT__ID f e <=> (!x. happ (happ f x) e = x)
% 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_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_lists_FOLDL2u_u_defu_c0]: !f cs c bs b a. h4/list/FOLDL2 f a (happ (happ h4/list/CONS b) bs) (happ (happ h4/list/CONS c) cs) = h4/list/FOLDL2 f (happ (happ (happ f a) b) c) bs cs
% Assm [h4s_lists_FOLDL2u_u_defu_c1]: !f cs a. h4/list/FOLDL2 f a h4/list/NIL cs = a
% Assm [h4s_lists_LENGTHu_u_NILu_u_SYM]: !l. h4/num/0 = h4/list/LENGTH l <=> l = h4/list/NIL
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (happ (happ h4/list/CONS h) t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_predu_u_sets_FINITEu_u_COMPLETEu_u_INDUCTION]: !P. (!x. (!y. h4/pred__set/PSUBSET y x ==> happ P y) ==> h4/pred__set/FINITE x ==> happ P x) ==> (!x. h4/pred__set/FINITE x ==> happ P x)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_predu_u_sets_ITSETu_u_THM]: !s f b. h4/pred__set/FINITE s ==> (?v. (v <=> s = h4/pred__set/EMPTY) /\ h4/pred__set/ITSET f s b = h4/bool/COND v b (h4/pred__set/ITSET f (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b)))
% Assm [h4s_predu_u_sets_FINITEu_u_EMPTY]: h4/pred__set/FINITE h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_RESTu_u_PSUBSET]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/PSUBSET (h4/pred__set/REST s) s
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (happ (h4/combin/C f) x) y = happ (happ f y) x
% Assm [h4s_predu_u_sets_FINITEu_u_REST]: !s. h4/pred__set/FINITE s ==> h4/pred__set/FINITE (h4/pred__set/REST s)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lists_SETu_u_TOu_u_LISTu_u_THM]: !s. h4/pred__set/FINITE s ==> (?v. (v <=> s = h4/pred__set/EMPTY) /\ h4/list/SET__TO__LIST s = h4/bool/COND v h4/list/NIL (happ (happ h4/list/CONS (h4/pred__set/CHOICE s)) (h4/list/SET__TO__LIST (h4/pred__set/REST s))))
% 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_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_lists_LUPDATEu_u_defu_c2]: !x n l e. h4/list/LUPDATE e (h4/num/SUC n) (happ (happ h4/list/CONS x) l) = happ (happ h4/list/CONS x) (h4/list/LUPDATE e n l)
% Assm [h4s_lists_LUPDATEu_u_defu_c1]: !x l e. h4/list/LUPDATE e h4/num/0 (happ (happ h4/list/CONS x) l) = happ (happ h4/list/CONS e) l
% Assm [h4s_lists_LUPDATEu_u_defu_c0]: !n e. h4/list/LUPDATE e n h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_EVERYu_u_DEFu_c1]: !t h P. h4/list/EVERY P (happ (happ h4/list/CONS h) t) <=> happ P h /\ h4/list/EVERY P t
% Assm [h4s_lists_EVERYu_u_DEFu_c0]: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_lists_MEMu_c0]: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm [h4s_lists_ALLu_u_DISTINCT0u_c0]: h4/list/ALL__DISTINCT h4/list/NIL <=> T
% Assm [h4s_lists_MEMu_u_MAP]: !x l f. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/MAP f l)) <=> (?y. x = happ f y /\ h4/bool/IN y (h4/list/LIST__TO__SET l))
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_lists_ALLu_u_DISTINCT0u_c1]: !t h. h4/list/ALL__DISTINCT (happ (happ h4/list/CONS h) t) <=> ~h4/bool/IN h (h4/list/LIST__TO__SET t) /\ h4/list/ALL__DISTINCT t
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Goal: !_1. (!f l_27 x. happ (happ (happ _1 f) l_27) x = h4/list/SNOC (happ f x) l_27) ==> (!_0. (!f l_27. happ (happ _0 f) l_27 = happ (happ _1 f) l_27) ==> (!l f. h4/list/MAP f l = h4/list/FOLDL (happ _0 f) h4/list/NIL l))
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_Q1246101,TV_Q1246097]: ![V_f, V_g]: (![V_x]: s(TV_Q1246097,happ(s(t_fun(TV_Q1246101,TV_Q1246097),V_f),s(TV_Q1246101,V_x))) = s(TV_Q1246097,happ(s(t_fun(TV_Q1246101,TV_Q1246097),V_g),s(TV_Q1246101,V_x))) => s(t_fun(TV_Q1246101,TV_Q1246097),V_f) = s(t_fun(TV_Q1246101,TV_Q1246097),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_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_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_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_lists_FOLDLu_u_SNOC, 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),h4s_lists_snoc(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_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,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),V_l))))),s(TV_u_27a,V_x)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_richu_u_lists_SUFFIXu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_P, V_lu_27, V_x]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_lu_27))),s(TV_u_27a,V_x))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_lu_27))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) => ![V_uu_0]: (![V_P, V_lu_27]: s(t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_lu_27))) = s(t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_lu_27))) => ![V_l, V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_suffix(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_foldl(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27a)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_MAPu_u_SNOC, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f]: s(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_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_snoc(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(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_l)))))).
fof(ah4s_lists_SUMu_u_MAPu_u_FOLDL, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_a, V_f, V_e]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_h4s_nums_num))),V_uu_1),s(t_h4s_nums_num,V_a))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(TV_u_27a,V_e))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_e))))) => ![V_uu_0]: (![V_f, V_a]: s(t_fun(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_nums_num)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_nums_num))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_h4s_nums_num,V_a))) = s(t_fun(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_h4s_nums_num))),V_uu_1),s(t_h4s_nums_num,V_a))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))) => ![V_ls, V_f]: s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_map(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),V_ls))))) = s(t_h4s_nums_num,h4s_lists_foldl(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_nums_num)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_nums_num))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_ls)))))).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(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_FOLDLu_u_UNIONu_u_BIGUNION, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_1]: (![V_s0, V_f, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_s0))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27b,t_bool),V_s0),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_f, V_s0]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f))),s(t_fun(TV_u_27b,t_bool),V_s0))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_s0))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f))) => ![V_s, V_ls, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_lists_foldl(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f))),s(t_fun(TV_u_27b,t_bool),V_s),s(t_h4s_lists_list(TV_u_27a),V_ls))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27b,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_ls)))))))))))).
fof(ah4s_lists_FOLDLu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_lu_27, V_l, V_fu_27, V_f, V_bu_27, V_b]: ((s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),V_lu_27) & (s(TV_u_27b,V_b) = s(TV_u_27b,V_bu_27) & ![V_x, V_a]: (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_lu_27)))))) => 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_a))),s(TV_u_27a,V_x))) = 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_fu_27),s(TV_u_27b,V_a))),s(TV_u_27a,V_x)))))) => 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_b),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_fu_27),s(TV_u_27b,V_bu_27),s(t_h4s_lists_list(TV_u_27a),V_lu_27))))).
fof(ah4s_lists_FOLDLu_u_ZIPu_u_SAME, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_1]: (![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_27b,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b))),V_uu_1),s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),V_f))),s(TV_u_27b,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),V_f),s(TV_u_27b,V_x))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(TV_u_27a,V_y))))) => ![V_uu_0]: (![V_f, V_x]: s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b))),V_uu_0),s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),V_f))),s(TV_u_27b,V_x))) = s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b))),V_uu_1),s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),V_f))),s(TV_u_27b,V_x))) => ![V_ls, V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_ls),s(t_h4s_lists_list(TV_u_27a),V_ls))))))) = s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b))),V_uu_0),s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),TV_u_27b)),V_f))),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),V_ls)))))).
fof(ah4s_lists_FOLDL2u_u_FOLDL, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_f, V_a0]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(TV_u_27c,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27c,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_f))),s(TV_u_27c,V_a0))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_f),s(TV_u_27c,V_a0))))) => ![V_l2, V_l1]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l2))) => ![V_f, V_a]: s(TV_u_27c,h4s_lists_foldl2(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_f),s(TV_u_27c,V_a),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))) = s(TV_u_27c,h4s_lists_foldl(s(t_fun(TV_u_27c,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27c,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_f))),s(TV_u_27c,V_a),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2)))))))))).
fof(ah4s_lists_FOLDLu_u_UNIONu_u_BIGUNIONu_u_paired, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_s0, V_f, V_x, V_y]: s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool)))),happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))))),V_uu_2),s(t_fun(TV_u_27c,t_bool),V_s0))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),V_f))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27c,t_bool),V_s0),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) => ![V_uu_1]: (![V_s0, V_f, V_x]: s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool)))),happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))))),V_uu_1),s(t_fun(TV_u_27c,t_bool),V_s0))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),V_f))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool)))),happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))))),V_uu_2),s(t_fun(TV_u_27c,t_bool),V_s0))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),V_f))),s(TV_u_27a,V_x))) => ![V_uu_0]: (![V_f, V_s0]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(t_fun(TV_u_27c,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),V_f))),s(t_fun(TV_u_27c,t_bool),V_s0))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool)))),happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))))),V_uu_1),s(t_fun(TV_u_27c,t_bool),V_s0))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),V_f))))) => ![V_s, V_ls, V_f]: s(t_fun(TV_u_27c,t_bool),h4s_lists_foldl(s(t_fun(t_fun(TV_u_27c,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),t_fun(t_fun(TV_u_27c,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),V_f))),s(t_fun(TV_u_27c,t_bool),V_s),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls))) = s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27c,t_bool),V_s),s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27c,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,t_bool))),V_f))),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_ls))))))))))))).
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_richu_u_lists_LENGTHu_u_FOLDL, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_lu_27, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_lu_27))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_lu_27))) => ![V_uu_0]: (![V_lu_27]: s(t_fun(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_nums_num)),V_uu_0),s(t_h4s_nums_num,V_lu_27))) = s(t_fun(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_lu_27))) => ![V_l]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_nums_num,h4s_lists_foldl(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_nums_num)),V_uu_0),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_richu_u_lists_FOLDRu_u_FOLDL, 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)))) => ![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))) = 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))))).
fof(ah4s_lists_ITSETu_u_equ_u_FOLDLu_u_SETu_u_TOu_u_LIST, axiom, ![TV_u_27b,TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f, V_a]: s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27b,V_a))) = s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(TV_u_27b,V_a),s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s))))))).
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),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),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_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_richu_u_lists_MAPu_u_FOLDR, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_1]: (![V_f, V_x, V_lu_27]: 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_27a,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27b),V_lu_27))) = 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_x))))),s(t_h4s_lists_list(TV_u_27b),V_lu_27))) => ![V_uu_0]: (![V_f, V_x]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))) => ![V_l, V_f]: s(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_l))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_EXISTSu_u_LISTu_u_EQu_u_MAP, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_f, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(TV_u_27a,V_x)))) <=> ?[V_y]: s(TV_u_27a,V_x) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(TV_u_27b,V_y)))) => ![V_ls, V_f]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_lists_list(TV_u_27a),V_ls)))) => ?[V_l]: s(t_h4s_lists_list(TV_u_27a),V_ls) = s(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),V_l)))))).
fof(ah4s_lists_LUPDATEu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_n, V_l, V_f]: s(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_lupdate(s(TV_u_27a,V_x),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_lupdate(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n),s(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_l)))))).
fof(ah4s_lists_ALLu_u_DISTINCTu_u_MAPu_u_INJ, axiom, ![TV_u_27b,TV_u_27a]: ![V_ls, V_f]: ((![V_x, V_y]: ((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_ls)))))) & (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_ls)))))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) & p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_ls))))) => p(s(t_bool,h4s_lists_allu_u_distinct(s(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_ls)))))))).
fof(ah4s_lists_SNOCu_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_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)))) => ![V_x]: 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_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l)))))))) => ![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_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_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_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_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_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_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_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_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_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_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_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_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_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_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_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_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_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_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_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_SNOCu_u_APPEND, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))))).
fof(ah4s_lists_MAPu_u_APPEND, axiom, ![TV_u_27b,TV_u_27a]: ![V_l2, V_l1, V_f]: s(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_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_27b),h4s_lists_append(s(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_l1))),s(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_l2)))))).
fof(ah4s_lists_ZIP0u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x2, V_x1, V_l2, V_l1]: s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(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_x1))),s(t_h4s_lists_list(TV_u_27a),V_l1))),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_x2))),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),h4s_lists_cons),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27b,V_x2))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2)))))))).
fof(ah4s_lists_ZIP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil)).
fof(ah4s_lists_SNOC0u_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(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_xu_27))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_xu_27))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_SNOC0u_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(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),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)))).
fof(ah4s_lists_listu_u_induction, 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_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_UNIONu_u_ASSOC, axiom, ![TV_u_27a]: ![V_u, V_t, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(TV_u_27a,t_bool),V_u)))).
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_BIGUNIONu_u_EMPTY, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_IMAGEu_u_EMPTY, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_UNIONu_u_EMPTYu_c1, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_predu_u_sets_IMAGEu_u_INSERT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_s, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_BIGUNIONu_u_INSERT, axiom, ![TV_u_27a]: ![V_s, V_P]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_insert(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
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_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_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_LENGTHu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),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_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_x))),s(TV_u_27b,V_y)))).
fof(ah4s_lists_SUM0u_c0, axiom, s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_lists_SUMu_u_SNOC, axiom, ![V_x, V_l]: s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_snoc(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(t_h4s_nums_num),V_l))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),V_l))),s(t_h4s_nums_num,V_x)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_operators_LEFTu_u_IDu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: (p(s(t_bool,h4s_operators_leftu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,V_e)))) <=> ![V_x]: 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_e))),s(TV_u_27b,V_x))) = s(TV_u_27b,V_x))).
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_operators_RIGHTu_u_IDu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: (p(s(t_bool,h4s_operators_rightu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27b,V_e)))) <=> ![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_x))),s(TV_u_27b,V_e))) = s(TV_u_27a,V_x))).
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_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) => p(s(t_bool,V_t))) <=> 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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_FOLDL2u_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f, V_cs, V_c, V_bs, V_b, V_a]: s(TV_u_27a,h4s_lists_foldl2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a))),V_f),s(TV_u_27a,V_a),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_b))),s(t_h4s_lists_list(TV_u_27b),V_bs))),s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(TV_u_27c,t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c))),h4s_lists_cons),s(TV_u_27c,V_c))),s(t_h4s_lists_list(TV_u_27c),V_cs))))) = s(TV_u_27a,h4s_lists_foldl2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a))),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a))),V_f),s(TV_u_27a,V_a))),s(TV_u_27b,V_b))),s(TV_u_27c,V_c))),s(t_h4s_lists_list(TV_u_27b),V_bs),s(t_h4s_lists_list(TV_u_27c),V_cs)))).
fof(ah4s_lists_FOLDL2u_u_defu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_f, V_cs, V_a]: s(TV_u_27a,h4s_lists_foldl2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a))),V_f),s(TV_u_27a,V_a),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27c),V_cs))) = s(TV_u_27a,V_a)).
fof(ah4s_lists_LENGTHu_u_NILu_u_SYM, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))) <=> s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(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_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_predu_u_sets_FINITEu_u_COMPLETEu_u_INDUCTION, axiom, ![TV_u_27a]: ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_y),s(t_fun(TV_u_27a,t_bool),V_x)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_y))))) => (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_x)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_x)))))) => ![V_x]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_x)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_x))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_predu_u_sets_ITSETu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_f, V_b]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27b,V_b))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),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,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b)))))))))).
fof(ah4s_predu_u_sets_FINITEu_u_EMPTY, axiom, ![TV_u_27a]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))).
fof(ah4s_predu_u_sets_RESTu_u_PSUBSET, axiom, ![TV_u_27a]: ![V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s)))))).
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),happ(s(t_fun(TV_u_27b,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_predu_u_sets_FINITEu_u_REST, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lists_SETu_u_TOu_u_LISTu_u_THM, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),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,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s)))))))))))).
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_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))).
fof(ah4s_lists_LUPDATEu_u_defu_c2, axiom, ![TV_u_27a]: ![V_x, V_n, V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),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(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_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_LUPDATEu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,h4s_nums_0),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(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_e))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_lists_LUPDATEu_u_defu_c0, axiom, ![TV_u_27a]: ![V_n, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,V_n),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_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_EVERYu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
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,f0)).
fof(ah4s_lists_ALLu_u_DISTINCT0u_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_lists_MEMu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(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_l)))))))) <=> ?[V_y]: (s(TV_u_27b,V_x) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l))))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_ALLu_u_DISTINCT0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: (p(s(t_bool,h4s_lists_allu_u_distinct(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)))))) <=> (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_h),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))))))) & p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_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_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> ?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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(ch4s_richu_u_lists_MAPu_u_FOLDL, conjecture, ![TV_u_27b,TV_u_27a]: ![V_uu_1]: (![V_f, V_lu_27, V_x]: s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(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_27b),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27b),V_lu_27))),s(TV_u_27a,V_x))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_snoc(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27b),V_lu_27))) => ![V_uu_0]: (![V_f, V_lu_27]: s(t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(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_27b),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27b),V_lu_27))) = s(t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(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_27b),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27b),V_lu_27))) => ![V_l, V_f]: s(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_l))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_foldl(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(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_27b),t_fun(TV_u_27a,t_h4s_lists_list(TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
