%   ORIGINAL: h4/fmapal/o__f__fmap
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/fmapal/o__f__bt__map: !t f cmp. h4/finite__map/o__f f (h4/fmapal/FMAPAL cmp t) = h4/fmapal/FMAPAL cmp (h4/fmapal/bt__map (h4/fmapal/AP__SND f) t)
% Assm: h4/fmapal/AP__SND0: !f b a. h4/fmapal/AP__SND f (h4/pair/_2C a b) = h4/pair/_2C a (f b)
% Assm: h4/finite__map/fmap__rel__FUPDATE__LIST__same: !ls2 ls1 f2 f1 R. h4/finite__map/fmap__rel R f1 f2 /\ h4/list/MAP h4/pair/FST ls1 = h4/list/MAP h4/pair/FST ls2 /\ h4/list/LIST__REL R (h4/list/MAP h4/pair/SND ls1) (h4/list/MAP h4/pair/SND ls2) ==> h4/finite__map/fmap__rel R (h4/finite__map/FUPDATE__LIST f1 ls1) (h4/finite__map/FUPDATE__LIST f2 ls2)
% 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/finite__map/o__f__FDOM: !g f. h4/finite__map/FDOM g = h4/finite__map/FDOM (h4/finite__map/o__f f g)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/TRUTH: T
% Assm: h4/finite__map/o__f__DEF_c0: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/finite__map/FDOM__o__f: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/finite__map/o__f__DEF_c1: !x g f. h4/bool/IN x (h4/finite__map/FDOM (h4/finite__map/o__f f g)) ==> h4/finite__map/FAPPLY (h4/finite__map/o__f f g) x = f (h4/finite__map/FAPPLY g x)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/finite__map/FEVERY__o__f: !m f P. h4/finite__map/FEVERY P (h4/finite__map/o__f f m) <=> h4/finite__map/FEVERY (\x. P (h4/pair/_2C (h4/pair/FST x) (f (h4/pair/SND x)))) m
% Assm: h4/finite__map/IMAGE__FRANGE: !fm f. h4/pred__set/IMAGE f (h4/finite__map/FRANGE fm) = h4/finite__map/FRANGE (h4/finite__map/o__f f fm)
% Assm: h4/finite__map/o__f__FUNION: !f2 f1 f. h4/finite__map/o__f f (h4/finite__map/FUNION f1 f2) = h4/finite__map/FUNION (h4/finite__map/o__f f f1) (h4/finite__map/o__f f f2)
% Assm: h4/finite__map/o__f__FAPPLY: !x g f. h4/bool/IN x (h4/finite__map/FDOM g) ==> h4/finite__map/FAPPLY (h4/finite__map/o__f f g) x = f (h4/finite__map/FAPPLY g x)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/enumeral/bt__induction: !P. P h4/enumeral/nt /\ (!b b0. P b /\ P b0 ==> (!a. P (h4/enumeral/node b a b0))) ==> (!b. P b)
% Assm: h4/fmapal/FAPPLY__node: !x r l cmp b a. h4/finite__map/FAPPLY (h4/fmapal/FMAPAL cmp (h4/enumeral/node l (h4/pair/_2C a b) r)) x = h4/toto/cpn__CASE (h4/toto/apto cmp x a) (h4/finite__map/FAPPLY (h4/fmapal/FMAPAL cmp l) x) b (h4/finite__map/FAPPLY (h4/fmapal/FMAPAL cmp r) x)
% Assm: h4/toto/cpn__case__def_c2: !v2 v1 v0. h4/toto/cpn__CASE h4/toto/GREATER v0 v1 v2 = v2
% Assm: h4/toto/cpn__case__def_c1: !v2 v1 v0. h4/toto/cpn__CASE h4/toto/EQUAL v0 v1 v2 = v1
% Assm: h4/toto/cpn__case__def_c0: !v2 v1 v0. h4/toto/cpn__CASE h4/toto/LESS v0 v1 v2 = v0
% Assm: h4/enumeral/bt__11: !a2_27 a2 a1_27 a1 a0_27 a0. h4/enumeral/node a0 a1 a2 = h4/enumeral/node a0_27 a1_27 a2_27 <=> a0 = a0_27 /\ a1 = a1_27 /\ a2 = a2_27
% Assm: h4/fmapal/bt__map0_c1: !x r l f. h4/fmapal/bt__map f (h4/enumeral/node l x r) = h4/enumeral/node (h4/fmapal/bt__map f l) (f x) (h4/fmapal/bt__map f r)
% Assm: h4/finite__map/fmap__EXT: !g f. f = g <=> h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x)
% Assm: h4/fmapal/bt__map0_c0: !f. h4/fmapal/bt__map f h4/enumeral/nt = h4/enumeral/nt
% Assm: h4/enumeral/bt__to__set_c0: !cmp. h4/enumeral/ENUMERAL cmp h4/enumeral/nt = h4/pred__set/EMPTY
% Assm: h4/toto/cpn__nchotomy: !a. a = h4/toto/LESS \/ a = h4/toto/EQUAL \/ a = h4/toto/GREATER
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/pair/PFORALL__THM: !P. (!x y. P x y) <=> $forall (h4/pair/UNCURRY (\x y. P x y))
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/fmapal/FMAPAL__FDOM__THM_c1: !x r l cmp b a. h4/bool/IN x (h4/finite__map/FDOM (h4/fmapal/FMAPAL cmp (h4/enumeral/node l (h4/pair/_2C a b) r))) <=> h4/toto/cpn__CASE (h4/toto/apto cmp x a) (h4/bool/IN x (h4/finite__map/FDOM (h4/fmapal/FMAPAL cmp l))) T (h4/bool/IN x (h4/finite__map/FDOM (h4/fmapal/FMAPAL cmp r)))
% Assm: h4/fmapal/bt__FST__FDOM: !t cmp. h4/finite__map/FDOM (h4/fmapal/FMAPAL cmp t) = h4/enumeral/ENUMERAL cmp (h4/fmapal/bt__map h4/pair/FST t)
% 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__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% 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/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__EMPTY: h4/pred__set/BIGUNION h4/pred__set/EMPTY = h4/pred__set/EMPTY
% 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/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/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/list/LIST__REL__def_c0: !R. h4/list/LIST__REL R h4/list/NIL h4/list/NIL <=> T
% Assm: h4/finite__map/fmap__rel__FUPDATE__same: !v2 v1 k f2 f1 R. h4/finite__map/fmap__rel R f1 f2 /\ R v1 v2 ==> h4/finite__map/fmap__rel R (h4/finite__map/FUPDATE f1 (h4/pair/_2C k v1)) (h4/finite__map/FUPDATE f2 (h4/pair/_2C k v2))
% Assm: h4/finite__map/FUPDATE__LIST__THM_c1: !t h f. h4/finite__map/FUPDATE__LIST f (h4/list/CONS h t) = h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE f h) t
% Assm: h4/finite__map/FUPDATE__LIST__THM_c0: !f. h4/finite__map/FUPDATE__LIST f h4/list/NIL = f
% Assm: h4/list/LIST__REL__def_c2: !bs b R. h4/list/LIST__REL R h4/list/NIL (h4/list/CONS b bs) <=> F
% Assm: h4/list/NOT__NIL__CONS: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% 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__REL__CONS1: !xs t h R. h4/list/LIST__REL R (h4/list/CONS h t) xs <=> (?h_27 t_27. xs = h4/list/CONS h_27 t_27 /\ R h h_27 /\ h4/list/LIST__REL R t t_27)
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/finite__map/FEVERY__DEF: !f P. h4/finite__map/FEVERY P f <=> (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> P (h4/pair/_2C x (h4/finite__map/FAPPLY f 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/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/finite__map/FUNION__DEF_c1: !x g f. h4/finite__map/FAPPLY (h4/finite__map/FUNION f g) x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/finite__map/FAPPLY f x) (h4/finite__map/FAPPLY g x)
% Assm: h4/finite__map/FUNION__DEF_c0: !g f. h4/finite__map/FDOM (h4/finite__map/FUNION f g) = h4/pred__set/UNION (h4/finite__map/FDOM f) (h4/finite__map/FDOM g)
% Assm: h4/finite__map/fmap__EQ__THM: !g f. h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x) <=> f = g
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/finite__map/fmap__INDUCT: !P. P h4/finite__map/FEMPTY /\ (!f. P f ==> (!x y. ~h4/bool/IN x (h4/finite__map/FDOM f) ==> P (h4/finite__map/FUPDATE f (h4/pair/_2C x y)))) ==> (!f. P f)
% Assm: h4/finite__map/FAPPLY__FUPDATE__THM: !x f b a. h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) x = h4/bool/COND (x = a) b (h4/finite__map/FAPPLY f x)
% Assm: h4/finite__map/FDOM__FUPDATE: !f b a. h4/finite__map/FDOM (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) = h4/pred__set/INSERT a (h4/finite__map/FDOM f)
% Assm: h4/finite__map/FDOM__FEMPTY: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/finite__map/FRANGE__DEF: !f. h4/finite__map/FRANGE f = h4/pred__set/GSPEC (\y. h4/pair/_2C y (?x. h4/bool/IN x (h4/finite__map/FDOM f) /\ h4/finite__map/FAPPLY f x = y))
% Assm: h4/finite__map/o__f__FRANGE: !x g f. h4/bool/IN x (h4/finite__map/FRANGE g) ==> h4/bool/IN (f x) (h4/finite__map/FRANGE (h4/finite__map/o__f f g))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Goal: !l f. h4/finite__map/o__f f (h4/fmapal/fmap l) = h4/fmapal/fmap (h4/list/MAP (h4/fmapal/AP__SND f) 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_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_fmapals_ou_u_fu_u_btu_u_map]: !t f cmp. h4/finite__map/o__f f (h4/fmapal/FMAPAL cmp t) = h4/fmapal/FMAPAL cmp (h4/fmapal/bt__map (h4/fmapal/AP__SND f) t)
% Assm [h4s_fmapals_APu_u_SND0]: !f b a. happ (h4/fmapal/AP__SND f) (h4/pair/_2C a b) = h4/pair/_2C a (happ f b)
% Assm [h4s_finiteu_u_maps_fmapu_u_relu_u_FUPDATEu_u_LISTu_u_same]: !ls2 ls1 f2 f1 R. h4/finite__map/fmap__rel R f1 f2 /\ h4/list/MAP h4/pair/FST ls1 = h4/list/MAP h4/pair/FST ls2 /\ h4/list/LIST__REL R (h4/list/MAP h4/pair/SND ls1) (h4/list/MAP h4/pair/SND ls2) ==> h4/finite__map/fmap__rel R (h4/finite__map/FUPDATE__LIST f1 ls1) (h4/finite__map/FUPDATE__LIST f2 ls2)
% 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_finiteu_u_maps_ou_u_fu_u_FDOM]: !g f. h4/finite__map/FDOM g = h4/finite__map/FDOM (h4/finite__map/o__f f g)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_DEFu_c0]: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_finiteu_u_maps_FDOMu_u_ou_u_f]: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_DEFu_c1]: !x g f. h4/bool/IN x (h4/finite__map/FDOM (h4/finite__map/o__f f g)) ==> h4/finite__map/FAPPLY (h4/finite__map/o__f f g) x = happ f (h4/finite__map/FAPPLY g x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_finiteu_u_maps_FEVERYu_u_ou_u_f]: !_0. (!P f x. happ (happ (happ _0 P) f) x <=> happ P (h4/pair/_2C (happ h4/pair/FST x) (happ f (happ h4/pair/SND x)))) ==> (!m f P. h4/finite__map/FEVERY P (h4/finite__map/o__f f m) <=> h4/finite__map/FEVERY (happ (happ _0 P) f) m)
% Assm [h4s_finiteu_u_maps_IMAGEu_u_FRANGE]: !fm f. h4/pred__set/IMAGE f (h4/finite__map/FRANGE fm) = h4/finite__map/FRANGE (h4/finite__map/o__f f fm)
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_FUNION]: !f2 f1 f. h4/finite__map/o__f f (h4/finite__map/FUNION f1 f2) = h4/finite__map/FUNION (h4/finite__map/o__f f f1) (h4/finite__map/o__f f f2)
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_FAPPLY]: !x g f. h4/bool/IN x (h4/finite__map/FDOM g) ==> h4/finite__map/FAPPLY (h4/finite__map/o__f f g) x = happ f (h4/finite__map/FAPPLY g x)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_enumerals_btu_u_induction]: !P. happ P h4/enumeral/nt /\ (!b b0. happ P b /\ happ P b0 ==> (!a. happ P (h4/enumeral/node b a b0))) ==> (!b. happ P b)
% Assm [h4s_fmapals_FAPPLYu_u_node]: !x r l cmp b a. h4/finite__map/FAPPLY (h4/fmapal/FMAPAL cmp (h4/enumeral/node l (h4/pair/_2C a b) r)) x = h4/toto/cpn__CASE (h4/toto/apto cmp x a) (h4/finite__map/FAPPLY (h4/fmapal/FMAPAL cmp l) x) b (h4/finite__map/FAPPLY (h4/fmapal/FMAPAL cmp r) x)
% Assm [h4s_totos_cpnu_u_caseu_u_defu_c2]: !v2 v1 v0. h4/toto/cpn__CASE h4/toto/GREATER v0 v1 v2 = v2
% Assm [h4s_totos_cpnu_u_caseu_u_defu_c1]: !v2 v1 v0. h4/toto/cpn__CASE h4/toto/EQUAL v0 v1 v2 = v1
% Assm [h4s_totos_cpnu_u_caseu_u_defu_c0]: !v2 v1 v0. h4/toto/cpn__CASE h4/toto/LESS v0 v1 v2 = v0
% Assm [h4s_enumerals_btu_u_11]: !a2_27 a2 a1_27 a1 a0_27 a0. h4/enumeral/node a0 a1 a2 = h4/enumeral/node a0_27 a1_27 a2_27 <=> a0 = a0_27 /\ a1 = a1_27 /\ a2 = a2_27
% Assm [h4s_fmapals_btu_u_map0u_c1]: !x r l f. h4/fmapal/bt__map f (h4/enumeral/node l x r) = h4/enumeral/node (h4/fmapal/bt__map f l) (happ f x) (h4/fmapal/bt__map f r)
% Assm [h4s_finiteu_u_maps_fmapu_u_EXT]: !g f. f = g <=> h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x)
% Assm [h4s_fmapals_btu_u_map0u_c0]: !f. h4/fmapal/bt__map f h4/enumeral/nt = h4/enumeral/nt
% Assm [h4s_enumerals_btu_u_tou_u_setu_c0]: !cmp. h4/enumeral/ENUMERAL cmp h4/enumeral/nt = h4/pred__set/EMPTY
% Assm [h4s_totos_cpnu_u_nchotomy]: !a. a = h4/toto/LESS \/ a = h4/toto/EQUAL \/ a = h4/toto/GREATER
% Assm [h4s_combins_ou_u_THM]: !x g f. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_pairs_PFORALLu_u_THM]: !_1. (!P x y. happ (happ (happ _1 P) x) y <=> happ (happ P x) y) ==> (!_0. (!P x. happ (happ _0 P) x = happ (happ _1 P) x) ==> (!P. (!x y. happ (happ P x) y) <=> $forall (h4/pair/UNCURRY (happ _0 P))))
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (happ h4/pair/FST x) (happ h4/pair/SND x) = x
% Assm [h4s_fmapals_FMAPALu_u_FDOMu_u_THMu_c1]: !x r l cmp b a. h4/bool/IN x (h4/finite__map/FDOM (h4/fmapal/FMAPAL cmp (h4/enumeral/node l (h4/pair/_2C a b) r))) <=> h4/toto/cpn__CASE (h4/toto/apto cmp x a) (h4/bool/IN x (h4/finite__map/FDOM (h4/fmapal/FMAPAL cmp l))) T (h4/bool/IN x (h4/finite__map/FDOM (h4/fmapal/FMAPAL cmp r)))
% Assm [h4s_fmapals_btu_u_FSTu_u_FDOM]: !t cmp. h4/finite__map/FDOM (h4/fmapal/FMAPAL cmp t) = h4/enumeral/ENUMERAL cmp (h4/fmapal/bt__map h4/pair/FST t)
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c0]: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_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_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_lists_FOLDL0u_c0]: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_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 [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_EMPTY]: h4/pred__set/BIGUNION h4/pred__set/EMPTY = h4/pred__set/EMPTY
% 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_FOLDL0u_c1]: !x l f e. h4/list/FOLDL f e (h4/list/CONS x l) = h4/list/FOLDL f (happ (happ f e) x) l
% Assm [h4s_pairs_SND0]: !y x. happ h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c0]: !R. h4/list/LIST__REL R h4/list/NIL h4/list/NIL <=> T
% Assm [h4s_finiteu_u_maps_fmapu_u_relu_u_FUPDATEu_u_same]: !v2 v1 k f2 f1 R. h4/finite__map/fmap__rel R f1 f2 /\ happ (happ R v1) v2 ==> h4/finite__map/fmap__rel R (h4/finite__map/FUPDATE f1 (h4/pair/_2C k v1)) (h4/finite__map/FUPDATE f2 (h4/pair/_2C k v2))
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c1]: !t h f. h4/finite__map/FUPDATE__LIST f (h4/list/CONS h t) = h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE f h) t
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c0]: !f. h4/finite__map/FUPDATE__LIST f h4/list/NIL = f
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c2]: !bs b R. h4/list/LIST__REL R h4/list/NIL (h4/list/CONS b bs) <=> F
% Assm [h4s_lists_NOTu_u_NILu_u_CONS]: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_lists_LISTu_u_RELu_u_CONS1]: !xs t h R. h4/list/LIST__REL R (h4/list/CONS h t) xs <=> (?h_27 t_27. xs = h4/list/CONS h_27 t_27 /\ happ (happ R h) h_27 /\ h4/list/LIST__REL R t t_27)
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (happ f h) (h4/list/MAP f t)
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_finiteu_u_maps_FEVERYu_u_DEF]: !f P. h4/finite__map/FEVERY P f <=> (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> happ P (h4/pair/_2C x (h4/finite__map/FAPPLY f 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_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_finiteu_u_maps_FUNIONu_u_DEFu_c1]: !x g f. h4/finite__map/FAPPLY (h4/finite__map/FUNION f g) x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/finite__map/FAPPLY f x) (h4/finite__map/FAPPLY g x)
% Assm [h4s_finiteu_u_maps_FUNIONu_u_DEFu_c0]: !g f. h4/finite__map/FDOM (h4/finite__map/FUNION f g) = h4/pred__set/UNION (h4/finite__map/FDOM f) (h4/finite__map/FDOM g)
% Assm [h4s_finiteu_u_maps_fmapu_u_EQu_u_THM]: !g f. h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x) <=> f = g
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_finiteu_u_maps_fmapu_u_INDUCT]: !P. happ P h4/finite__map/FEMPTY /\ (!f. happ P f ==> (!x y. ~h4/bool/IN x (h4/finite__map/FDOM f) ==> happ P (h4/finite__map/FUPDATE f (h4/pair/_2C x y)))) ==> (!f. happ P f)
% Assm [h4s_finiteu_u_maps_FAPPLYu_u_FUPDATEu_u_THM]: !x f b a. ?v. (v <=> x = a) /\ h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) x = h4/bool/COND v b (h4/finite__map/FAPPLY f x)
% Assm [h4s_finiteu_u_maps_FDOMu_u_FUPDATE]: !f b a. h4/finite__map/FDOM (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) = h4/pred__set/INSERT a (h4/finite__map/FDOM f)
% Assm [h4s_finiteu_u_maps_FDOMu_u_FEMPTY]: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_finiteu_u_maps_FRANGEu_u_DEF]: !_0. (!f y. ?v. (v <=> (?x. h4/bool/IN x (h4/finite__map/FDOM f) /\ h4/finite__map/FAPPLY f x = y)) /\ happ (happ _0 f) y = h4/pair/_2C y v) ==> (!f. h4/finite__map/FRANGE f = h4/pred__set/GSPEC (happ _0 f))
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_FRANGE]: !x g f. h4/bool/IN x (h4/finite__map/FRANGE g) ==> h4/bool/IN (happ f x) (h4/finite__map/FRANGE (h4/finite__map/o__f f g))
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Goal: !l f. h4/finite__map/o__f f (h4/fmapal/fmap l) = h4/fmapal/fmap (h4/list/MAP (h4/fmapal/AP__SND f) 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_Q1312682,TV_Q1312678]: ![V_f, V_g]: (![V_x]: s(TV_Q1312678,happ(s(t_fun(TV_Q1312682,TV_Q1312678),V_f),s(TV_Q1312682,V_x))) = s(TV_Q1312678,happ(s(t_fun(TV_Q1312682,TV_Q1312678),V_g),s(TV_Q1312682,V_x))) => s(t_fun(TV_Q1312682,TV_Q1312678),V_f) = s(t_fun(TV_Q1312682,TV_Q1312678),V_g))).
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_fmapals_ou_u_fu_u_btu_u_map, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_t, V_f, V_cmp]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_fmapals_btu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_fmapals_apu_u_snd(s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))))).
fof(ah4s_fmapals_APu_u_SND0, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_f, V_b, V_a]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_fmapals_apu_u_snd(s(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_a),s(TV_u_27b,V_b))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(TV_u_27b,V_b)))))).
fof(ah4s_finiteu_u_maps_fmapu_u_relu_u_FUPDATEu_u_LISTu_u_same, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_ls2, V_ls1, V_f2, V_f1, V_R]: ((p(s(t_bool,h4s_finiteu_u_maps_fmapu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2)))) & (s(t_h4s_lists_list(TV_u_27c),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),TV_u_27c),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),V_ls1))) = s(t_h4s_lists_list(TV_u_27c),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),TV_u_27c),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),V_ls2))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),TV_u_27a),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),V_ls1))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),V_ls2)))))))) => p(s(t_bool,h4s_finiteu_u_maps_fmapu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),V_ls1))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),V_ls2)))))))).
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_finiteu_u_maps_ou_u_fu_u_FDOM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))) = s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))))).
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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_DEFu_c0, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))).
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_finiteu_u_maps_FDOMu_u_ou_u_f, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_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_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_finiteu_u_maps_ou_u_fu_u_DEFu_c1, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))))))) => s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))))).
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_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_finiteu_u_maps_FEVERYu_u_ou_u_f, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_P, V_f, V_x]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_bool),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_bool),V_P))),s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))))))) => ![V_m, V_f, V_P]: s(t_bool,h4s_finiteu_u_maps_fevery(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_m))))) = s(t_bool,h4s_finiteu_u_maps_fevery(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_bool),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_bool),V_P))),s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_m))))).
fof(ah4s_finiteu_u_maps_IMAGEu_u_FRANGE, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_fm, 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_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_fm))))) = s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_fm)))))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_FUNION, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_f2, V_f1, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_f1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_f2))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_f1))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_f2)))))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_FAPPLY, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))))) => s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),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_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_enumerals_btu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_bool),V_P),s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_nt)))) & ![V_b, V_b0]: ((p(s(t_bool,happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_bool),V_P),s(t_h4s_enumerals_bt(TV_u_27a),V_b)))) & p(s(t_bool,happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_bool),V_P),s(t_h4s_enumerals_bt(TV_u_27a),V_b0))))) => ![V_a]: p(s(t_bool,happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_bool),V_P),s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_node(s(t_h4s_enumerals_bt(TV_u_27a),V_b),s(TV_u_27a,V_a),s(t_h4s_enumerals_bt(TV_u_27a),V_b0)))))))) => ![V_b]: p(s(t_bool,happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_bool),V_P),s(t_h4s_enumerals_bt(TV_u_27a),V_b)))))).
fof(ah4s_fmapals_FAPPLYu_u_node, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_r, V_l, V_cmp, V_b, V_a]: s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_enumerals_node(s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_r))))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_totos_cpnu_u_case(s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(TV_u_27a,V_a))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))),s(TV_u_27a,V_x))),s(TV_u_27b,V_b),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_r))),s(TV_u_27a,V_x)))))).
fof(ah4s_totos_cpnu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v2, V_v1, V_v0]: s(TV_u_27a,h4s_totos_cpnu_u_case(s(t_h4s_totos_cpn,h4s_totos_greater),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))) = s(TV_u_27a,V_v2)).
fof(ah4s_totos_cpnu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v2, V_v1, V_v0]: s(TV_u_27a,h4s_totos_cpnu_u_case(s(t_h4s_totos_cpn,h4s_totos_equal),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))) = s(TV_u_27a,V_v1)).
fof(ah4s_totos_cpnu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v2, V_v1, V_v0]: s(TV_u_27a,h4s_totos_cpnu_u_case(s(t_h4s_totos_cpn,h4s_totos_less),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))) = s(TV_u_27a,V_v0)).
fof(ah4s_enumerals_btu_u_11, axiom, ![TV_u_27a]: ![V_a2u_27, V_a2, V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_node(s(t_h4s_enumerals_bt(TV_u_27a),V_a0),s(TV_u_27a,V_a1),s(t_h4s_enumerals_bt(TV_u_27a),V_a2))) = s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_node(s(t_h4s_enumerals_bt(TV_u_27a),V_a0u_27),s(TV_u_27a,V_a1u_27),s(t_h4s_enumerals_bt(TV_u_27a),V_a2u_27))) <=> (s(t_h4s_enumerals_bt(TV_u_27a),V_a0) = s(t_h4s_enumerals_bt(TV_u_27a),V_a0u_27) & (s(TV_u_27a,V_a1) = s(TV_u_27a,V_a1u_27) & s(t_h4s_enumerals_bt(TV_u_27a),V_a2) = s(t_h4s_enumerals_bt(TV_u_27a),V_a2u_27))))).
fof(ah4s_fmapals_btu_u_map0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_r, V_l, V_f]: s(t_h4s_enumerals_bt(TV_u_27b),h4s_fmapals_btu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_node(s(t_h4s_enumerals_bt(TV_u_27a),V_l),s(TV_u_27a,V_x),s(t_h4s_enumerals_bt(TV_u_27a),V_r))))) = s(t_h4s_enumerals_bt(TV_u_27b),h4s_enumerals_node(s(t_h4s_enumerals_bt(TV_u_27b),h4s_fmapals_btu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_enumerals_bt(TV_u_27a),V_l))),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_enumerals_bt(TV_u_27b),h4s_fmapals_btu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_enumerals_bt(TV_u_27a),V_r)))))).
fof(ah4s_finiteu_u_maps_fmapu_u_EXT, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g) <=> (s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) => s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_fmapals_btu_u_map0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_enumerals_bt(TV_u_27b),h4s_fmapals_btu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_nt))) = s(t_h4s_enumerals_bt(TV_u_27b),h4s_enumerals_nt)).
fof(ah4s_enumerals_btu_u_tou_u_setu_c0, axiom, ![TV_u_27a]: ![V_cmp]: s(t_fun(TV_u_27a,t_bool),h4s_enumerals_enumeral(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_nt))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_totos_cpnu_u_nchotomy, axiom, ![V_a]: (s(t_h4s_totos_cpn,V_a) = s(t_h4s_totos_cpn,h4s_totos_less) | (s(t_h4s_totos_cpn,V_a) = s(t_h4s_totos_cpn,h4s_totos_equal) | s(t_h4s_totos_cpn,V_a) = s(t_h4s_totos_cpn,h4s_totos_greater)))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_pairs_PFORALLu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x, V_y]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),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))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) => ![V_uu_0]: (![V_P, 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))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,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))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))) => ![V_P]: (![V_x, V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> p(s(t_bool,d_forall(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_uncurry(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))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P)))))))))))).
fof(ah4s_pairs_PAIR, axiom, ![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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_fmapals_FMAPALu_u_FDOMu_u_THMu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_l, V_cmp, V_b, V_a]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_enumerals_node(s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_r))))))))) = s(t_bool,h4s_totos_cpnu_u_case(s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(TV_u_27a,V_a))),s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))))),s(t_bool,t),s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_r)))))))))).
fof(ah4s_fmapals_btu_u_FSTu_u_FDOM, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_cmp]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmapal(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))))) = s(t_fun(TV_u_27a,t_bool),h4s_enumerals_enumeral(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(TV_u_27a),h4s_fmapals_btu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_enumerals_bt(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_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_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),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_FOLDL0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_lists_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),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_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_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_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_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),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_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R]: s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_finiteu_u_maps_fmapu_u_relu_u_FUPDATEu_u_same, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_v2, V_v1, V_k, V_f2, V_f1, V_R]: ((p(s(t_bool,h4s_finiteu_u_maps_fmapu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2)))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_v1))),s(TV_u_27b,V_v2))))) => p(s(t_bool,h4s_finiteu_u_maps_fmapu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1),s(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27c,V_k),s(TV_u_27a,V_v1))))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2),s(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27c,V_k),s(TV_u_27b,V_v2)))))))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_h, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(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),V_h),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_h))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_bs, V_b, V_R]: s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(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_bool,f0)).
fof(ah4s_lists_NOTu_u_NILu_u_CONS, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0u_27),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_lists_listu_u_nchotomy, axiom, ![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) | ?[V_h, V_t]: s(t_h4s_lists_list(TV_u_27a),V_l) = s(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))))).
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),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_RELu_u_CONS1, axiom, ![TV_u_27a,TV_u_27b]: ![V_xs, V_t, V_h, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(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),V_xs)))) <=> ?[V_hu_27, V_tu_27]: (s(t_h4s_lists_list(TV_u_27b),V_xs) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_hu_27),s(t_h4s_lists_list(TV_u_27b),V_tu_27))) & (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_h))),s(TV_u_27b,V_hu_27)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_t),s(t_h4s_lists_list(TV_u_27b),V_tu_27)))))))).
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),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),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_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_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f0))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_finiteu_u_maps_FEVERYu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_finiteu_u_maps_fevery(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),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,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,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_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_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_finiteu_u_maps_FUNIONu_u_DEFu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_finiteu_u_maps_FUNIONu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))))).
fof(ah4s_finiteu_u_maps_fmapu_u_EQu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: ((s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) => s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))) <=> s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_finiteu_u_maps_fmapu_u_INDUCT, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty)))) & ![V_f]: (p(s(t_bool,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),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_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),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))))))))))) => ![V_f]: p(s(t_bool,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))).
fof(ah4s_finiteu_u_maps_FAPPLYu_u_FUPDATEu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f, V_b, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_a)) & s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_FUPDATE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_b, V_a]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_FEMPTY, axiom, ![TV_u_27b,TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_finiteu_u_maps_FRANGEu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_f, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) & s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,V_y))) & s(t_h4s_pairs_prod(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27b,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27b,t_bool))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27b,t_bool),h4s_pairs_u_2c(s(TV_u_27b,V_y),s(t_bool,V_v)))) => ![V_f]: s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27b,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27b,t_bool))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_FRANGE, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_g)))))) => p(s(t_bool,h4s_bools_in(s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27c,t_bool),h4s_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_g)))))))))).
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_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_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_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),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)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
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_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ch4s_fmapals_ou_u_fu_u_fmap, conjecture, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_l, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_fmapals_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_fmapals_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_fmapals_apu_u_snd(s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l)))))).
