%   ORIGINAL: h4/alist/fmap__to__alist__preserves__FDOM
% 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/finite__map/MAP__KEYS__witness: h4/bool/LET (\m. !f fm. h4/finite__map/FDOM (m f fm) = h4/pred__set/IMAGE f (h4/finite__map/FDOM fm) /\ (h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> (!x. h4/bool/IN x (h4/finite__map/FDOM fm) ==> h4/finite__map/FAPPLY (m f fm) (f x) = h4/finite__map/FAPPLY fm x))) (\f fm. h4/bool/COND (h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV) (h4/finite__map/f__o__f fm (h4/finite__map/FUN__FMAP (h4/pred__set/LINV f (h4/finite__map/FDOM fm)) (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))) (h4/finite__map/FUN__FMAP h4/bool/ARB (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm))))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/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/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/finite__map/f__o__f__DEF_c0: !g f. h4/finite__map/FDOM (h4/finite__map/f__o__f f g) = h4/pred__set/INTER (h4/finite__map/FDOM g) (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (h4/finite__map/FAPPLY g x) (h4/finite__map/FDOM f))))
% Assm: h4/finite__map/FMAP__MAP2__def: !m f. h4/finite__map/FMAP__MAP2 f m = h4/finite__map/FUN__FMAP (\x. f (h4/pair/_2C x (h4/finite__map/FAPPLY m x))) (h4/finite__map/FDOM m)
% Assm: h4/finite__map/MAP__KEYS__def_c0: !fm f. h4/finite__map/FDOM (h4/finite__map/MAP__KEYS f fm) = h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)
% Assm: h4/finite__map/f__o__DEF: !g f. h4/finite__map/f__o f g = h4/finite__map/f__o__f f (h4/finite__map/FUN__FMAP g (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (g x) (h4/finite__map/FDOM f)))))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> 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/finite__map/MAP__KEYS__FUPDATE: !v k fm f. h4/pred__set/INJ f (h4/pred__set/INSERT k (h4/finite__map/FDOM fm)) h4/pred__set/UNIV ==> h4/finite__map/MAP__KEYS f (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) = h4/finite__map/FUPDATE (h4/finite__map/MAP__KEYS f fm) (h4/pair/_2C (f k) v)
% Assm: h4/finite__map/FDOM__f__o: !g f. h4/pred__set/FINITE (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (g x) (h4/finite__map/FDOM f)))) ==> h4/finite__map/FDOM (h4/finite__map/f__o f g) = h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (g x) (h4/finite__map/FDOM f)))
% Assm: h4/finite__map/FMAP__MAP2__THM_c0: !m f. h4/finite__map/FDOM (h4/finite__map/FMAP__MAP2 f m) = h4/finite__map/FDOM m
% Assm: h4/finite__map/FINITE__PRED__11: !g. (!x y. g x = g y <=> x = y) ==> (!f. h4/pred__set/FINITE (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (g x) (h4/finite__map/FDOM f)))))
% 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/finite__map/MAP__KEYS__using__LINV: !fm f. h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> h4/finite__map/MAP__KEYS f fm = h4/finite__map/f__o__f fm (h4/finite__map/FUN__FMAP (h4/pred__set/LINV f (h4/finite__map/FDOM fm)) (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/alist/alist__to__fmap__to__alist: !al. h4/alist/fmap__to__alist (h4/alist/alist__to__fmap al) = h4/list/MAP (\k. h4/pair/_2C k (h4/option/THE (h4/alist/ALOOKUP al k))) (h4/list/SET__TO__LIST (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)))
% Assm: h4/alist/ALL__DISTINCT__fmap__to__alist__keys: !fm. h4/list/ALL__DISTINCT (h4/list/MAP h4/pair/FST (h4/alist/fmap__to__alist fm))
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/finite__map/FUN__FMAP__DEF: !f P. h4/pred__set/FINITE P ==> h4/finite__map/FDOM (h4/finite__map/FUN__FMAP f P) = P /\ (!x. h4/bool/IN x P ==> h4/finite__map/FAPPLY (h4/finite__map/FUN__FMAP f P) x = f x)
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/finite__map/FDOM__FINITE: !fm. h4/pred__set/FINITE (h4/finite__map/FDOM fm)
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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/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/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/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% 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/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/FDOM__FEMPTY: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/finite__map/MAP__KEYS__def_c1: !fm f. h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> (!x. h4/bool/IN x (h4/finite__map/FDOM fm) ==> h4/finite__map/FAPPLY (h4/finite__map/MAP__KEYS f fm) (f x) = h4/finite__map/FAPPLY fm x)
% 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/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% 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/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> 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_c2: ~F <=> T
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/pred__set/GSPEC__OR: !Q P. h4/pred__set/GSPEC (\x. h4/pair/_2C x (P x \/ Q x)) = h4/pred__set/UNION (h4/pred__set/GSPEC (\x. h4/pair/_2C x (P x))) (h4/pred__set/GSPEC (\x. h4/pair/_2C x (Q x)))
% Assm: h4/pred__set/GSPEC__EQ: !y. h4/pred__set/GSPEC (\x. h4/pair/_2C x (x = y)) = h4/pred__set/INSERT y h4/pred__set/EMPTY
% Assm: h4/pred__set/GSPEC__F: h4/pred__set/GSPEC (\x. h4/pair/_2C x F) = h4/pred__set/EMPTY
% Assm: h4/pred__set/FINITE__UNION: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm: h4/pred__set/SING__FINITE: !s. h4/pred__set/SING s ==> h4/pred__set/FINITE s
% Assm: h4/pred__set/FINITE__EMPTY: h4/pred__set/FINITE h4/pred__set/EMPTY
% Assm: h4/pred__set/SING0: !x. h4/pred__set/SING (h4/pred__set/INSERT x h4/pred__set/EMPTY)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% 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/list/MAP__EQ__f: !l f2 f1. h4/list/MAP f1 l = h4/list/MAP f2 l <=> (!e. h4/bool/IN e (h4/list/LIST__TO__SET l) ==> f1 e = f2 e)
% Assm: h4/alist/fmap__to__alist__def: !s. h4/alist/fmap__to__alist s = h4/list/MAP (\k. h4/pair/_2C k (h4/finite__map/FAPPLY s k)) (h4/list/SET__TO__LIST (h4/finite__map/FDOM s))
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/list/MEM__SET__TO__LIST: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s)) <=> h4/bool/IN x s)
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/pred__set/IMAGE__FINITE: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm: h4/pred__set/IMAGE__IN: !x s. h4/bool/IN x s ==> (!f. h4/bool/IN (f x) (h4/pred__set/IMAGE f s))
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/pred__set/LINV__DEF: !t s f. h4/pred__set/INJ f s t ==> (!x. h4/bool/IN x s ==> h4/pred__set/LINV f s (f x) = x)
% Assm: h4/pred__set/SUBSET__INTER__ABSORPTION: !t s. h4/pred__set/SUBSET s t <=> h4/pred__set/INTER s t = s
% Assm: h4/finite__map/f__o__f__DEF_c1: !x g f. h4/bool/IN x (h4/finite__map/FDOM (h4/finite__map/f__o__f f g)) ==> h4/finite__map/FAPPLY (h4/finite__map/f__o__f f g) x = h4/finite__map/FAPPLY f (h4/finite__map/FAPPLY g x)
% Assm: h4/alist/FDOM__alist__to__fmap: !al. h4/finite__map/FDOM (h4/alist/alist__to__fmap al) = h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)
% Assm: h4/list/FINITE__LIST__TO__SET: !l. h4/pred__set/FINITE (h4/list/LIST__TO__SET l)
% Assm: h4/alist/ALOOKUP__SOME__FAPPLY__alist__to__fmap: !v k al. h4/alist/ALOOKUP al k = h4/option/SOME v ==> h4/finite__map/FAPPLY (h4/alist/alist__to__fmap al) k = v
% Assm: h4/alist/ALOOKUP__FAILS: !x l. h4/alist/ALOOKUP l x = h4/option/NONE <=> (!k v. h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET l) ==> ~(k = x))
% Assm: h4/pair/pair__CASES: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/pred__set/INJ__DEF: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> f x = f y ==> x = y)
% 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/INJ__INSERT: !x t s f. h4/pred__set/INJ f (h4/pred__set/INSERT x s) t <=> h4/pred__set/INJ f s t /\ h4/bool/IN (f x) t /\ (!y. h4/bool/IN y s /\ f x = f y ==> x = y)
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/pred__set/INTER__EMPTY_c0: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/COND__EXPAND: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% Assm: h4/bool/COND__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Assm: h4/bool/COND__RATOR: !x g f b. h4/bool/COND b f g x = h4/bool/COND b (f x) (g x)
% 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/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/list/SET__TO__LIST__primitive: h4/list/SET__TO__LIST = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!s. h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> R (h4/pred__set/REST s) s))) (\SET__TO__LIST s. h4/combin/I (h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND (s = h4/pred__set/EMPTY) h4/list/NIL (h4/list/CONS (h4/pred__set/CHOICE s) (SET__TO__LIST (h4/pred__set/REST s)))) h4/bool/ARB))
% Goal: !fm2 fm1. h4/finite__map/FDOM fm1 = h4/finite__map/FDOM fm2 ==> h4/list/MAP h4/pair/FST (h4/alist/fmap__to__alist fm1) = h4/list/MAP h4/pair/FST (h4/alist/fmap__to__alist fm2)
%   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_finiteu_u_maps_MAPu_u_KEYSu_u_witness]: !_2. (!f fm. happ (happ _2 f) fm = h4/bool/COND (h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV) (h4/finite__map/f__o__f fm (h4/finite__map/FUN__FMAP (h4/pred__set/LINV f (h4/finite__map/FDOM fm)) (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))) (h4/finite__map/FUN__FMAP h4/bool/ARB (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))) ==> (!_1. (!f. happ _1 f = happ _2 f) ==> (!_0. (!m. happ _0 m <=> (!f fm. h4/finite__map/FDOM (happ (happ m f) fm) = h4/pred__set/IMAGE f (h4/finite__map/FDOM fm) /\ (h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> (!x. h4/bool/IN x (h4/finite__map/FDOM fm) ==> h4/finite__map/FAPPLY (happ (happ m f) fm) (happ f x) = h4/finite__map/FAPPLY fm x)))) ==> h4/bool/LET _0 _1))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_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_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_finiteu_u_maps_fu_u_ou_u_fu_u_DEFu_c0]: !_0. (!g f x. happ (happ (happ _0 g) f) x = h4/pair/_2C x (h4/bool/IN (h4/finite__map/FAPPLY g x) (h4/finite__map/FDOM f))) ==> (!g f. h4/finite__map/FDOM (h4/finite__map/f__o__f f g) = h4/pred__set/INTER (h4/finite__map/FDOM g) (h4/pred__set/GSPEC (happ (happ _0 g) f)))
% Assm [h4s_finiteu_u_maps_FMAPu_u_MAP2u_u_def]: !_0. (!f m x. happ (happ (happ _0 f) m) x = happ f (h4/pair/_2C x (h4/finite__map/FAPPLY m x))) ==> (!m f. h4/finite__map/FMAP__MAP2 f m = h4/finite__map/FUN__FMAP (happ (happ _0 f) m) (h4/finite__map/FDOM m))
% Assm [h4s_finiteu_u_maps_MAPu_u_KEYSu_u_defu_c0]: !fm f. h4/finite__map/FDOM (h4/finite__map/MAP__KEYS f fm) = h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)
% Assm [h4s_finiteu_u_maps_fu_u_ou_u_DEF]: !_0. (!g f x. happ (happ (happ _0 g) f) x = h4/pair/_2C x (h4/bool/IN (happ g x) (h4/finite__map/FDOM f))) ==> (!g f. h4/finite__map/f__o f g = h4/finite__map/f__o__f f (h4/finite__map/FUN__FMAP g (h4/pred__set/GSPEC (happ (happ _0 g) f))))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% 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_finiteu_u_maps_MAPu_u_KEYSu_u_FUPDATE]: !v k fm f. h4/pred__set/INJ f (h4/pred__set/INSERT k (h4/finite__map/FDOM fm)) h4/pred__set/UNIV ==> h4/finite__map/MAP__KEYS f (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) = h4/finite__map/FUPDATE (h4/finite__map/MAP__KEYS f fm) (h4/pair/_2C (happ f k) v)
% Assm [h4s_finiteu_u_maps_FDOMu_u_fu_u_o]: !_0. (!g f x. happ (happ (happ _0 g) f) x = h4/pair/_2C x (h4/bool/IN (happ g x) (h4/finite__map/FDOM f))) ==> (!g f. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 g) f)) ==> h4/finite__map/FDOM (h4/finite__map/f__o f g) = h4/pred__set/GSPEC (happ (happ _0 g) f))
% Assm [h4s_finiteu_u_maps_FMAPu_u_MAP2u_u_THMu_c0]: !m f. h4/finite__map/FDOM (h4/finite__map/FMAP__MAP2 f m) = h4/finite__map/FDOM m
% Assm [h4s_finiteu_u_maps_FINITEu_u_PREDu_u_11]: !_0. (!g f x. happ (happ (happ _0 g) f) x = h4/pair/_2C x (h4/bool/IN (happ g x) (h4/finite__map/FDOM f))) ==> (!g. (!x y. happ g x = happ g y <=> x = y) ==> (!f. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 g) f))))
% 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_finiteu_u_maps_MAPu_u_KEYSu_u_usingu_u_LINV]: !fm f. h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> h4/finite__map/MAP__KEYS f fm = h4/finite__map/f__o__f fm (h4/finite__map/FUN__FMAP (h4/pred__set/LINV f (h4/finite__map/FDOM fm)) (h4/pred__set/IMAGE f (h4/finite__map/FDOM fm)))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_alists_alistu_u_tou_u_fmapu_u_tou_u_alist]: !_0. (!al k. happ (happ _0 al) k = h4/pair/_2C k (h4/option/THE (h4/alist/ALOOKUP al k))) ==> (!al. h4/alist/fmap__to__alist (h4/alist/alist__to__fmap al) = h4/list/MAP (happ _0 al) (happ h4/list/SET__TO__LIST (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al))))
% Assm [h4s_alists_ALLu_u_DISTINCTu_u_fmapu_u_tou_u_alistu_u_keys]: !fm. h4/list/ALL__DISTINCT (h4/list/MAP h4/pair/FST (h4/alist/fmap__to__alist fm))
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_finiteu_u_maps_FUNu_u_FMAPu_u_DEF]: !f P. h4/pred__set/FINITE P ==> h4/finite__map/FDOM (h4/finite__map/FUN__FMAP f P) = P /\ (!x. h4/bool/IN x P ==> h4/finite__map/FAPPLY (h4/finite__map/FUN__FMAP f P) x = happ f x)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_finiteu_u_maps_FDOMu_u_FINITE]: !fm. h4/pred__set/FINITE (h4/finite__map/FDOM fm)
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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_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_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_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% 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_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_FDOMu_u_FEMPTY]: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_finiteu_u_maps_MAPu_u_KEYSu_u_defu_c1]: !fm f. h4/pred__set/INJ f (h4/finite__map/FDOM fm) h4/pred__set/UNIV ==> (!x. h4/bool/IN x (h4/finite__map/FDOM fm) ==> h4/finite__map/FAPPLY (h4/finite__map/MAP__KEYS f fm) (happ f x) = h4/finite__map/FAPPLY fm x)
% 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_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% 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_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_predu_u_sets_GSPECu_u_OR]: !_1. (!P x. happ (happ _1 P) x = h4/pair/_2C x (happ P x)) ==> (!_0. (!P Q x. ?v. (v <=> happ P x \/ happ Q x) /\ happ (happ (happ _0 P) Q) x = h4/pair/_2C x v) ==> (!Q P. h4/pred__set/GSPEC (happ (happ _0 P) Q) = h4/pred__set/UNION (h4/pred__set/GSPEC (happ _1 P)) (h4/pred__set/GSPEC (happ _1 Q))))
% Assm [h4s_predu_u_sets_GSPECu_u_EQ]: !_0. (!y x. ?v. (v <=> x = y) /\ happ (happ _0 y) x = h4/pair/_2C x v) ==> (!y. h4/pred__set/GSPEC (happ _0 y) = h4/pred__set/INSERT y h4/pred__set/EMPTY)
% Assm [h4s_predu_u_sets_GSPECu_u_F]: !_0. (!x. happ _0 x = h4/pair/_2C x F) ==> h4/pred__set/GSPEC _0 = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_FINITEu_u_UNION]: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm [h4s_predu_u_sets_SINGu_u_FINITE]: !s. h4/pred__set/SING s ==> h4/pred__set/FINITE s
% Assm [h4s_predu_u_sets_FINITEu_u_EMPTY]: h4/pred__set/FINITE h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_SING0]: !x. h4/pred__set/SING (h4/pred__set/INSERT x h4/pred__set/EMPTY)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% 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_lists_MAPu_u_EQu_u_f]: !l f2 f1. h4/list/MAP f1 l = h4/list/MAP f2 l <=> (!e. h4/bool/IN e (h4/list/LIST__TO__SET l) ==> happ f1 e = happ f2 e)
% Assm [h4s_alists_fmapu_u_tou_u_alistu_u_def]: !_0. (!s k. happ (happ _0 s) k = h4/pair/_2C k (h4/finite__map/FAPPLY s k)) ==> (!s. h4/alist/fmap__to__alist s = h4/list/MAP (happ _0 s) (happ h4/list/SET__TO__LIST (h4/finite__map/FDOM s)))
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_lists_MEMu_u_SETu_u_TOu_u_LIST]: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x (h4/list/LIST__TO__SET (happ h4/list/SET__TO__LIST s)) <=> h4/bool/IN x s)
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_predu_u_sets_IMAGEu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm [h4s_predu_u_sets_IMAGEu_u_IN]: !x s. h4/bool/IN x s ==> (!f. h4/bool/IN (happ f x) (h4/pred__set/IMAGE f s))
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_LINVu_u_DEF]: !t s f. h4/pred__set/INJ f s t ==> (!x. h4/bool/IN x s ==> happ (h4/pred__set/LINV f s) (happ f x) = x)
% Assm [h4s_predu_u_sets_SUBSETu_u_INTERu_u_ABSORPTION]: !t s. h4/pred__set/SUBSET s t <=> h4/pred__set/INTER s t = s
% Assm [h4s_finiteu_u_maps_fu_u_ou_u_fu_u_DEFu_c1]: !x g f. h4/bool/IN x (h4/finite__map/FDOM (h4/finite__map/f__o__f f g)) ==> h4/finite__map/FAPPLY (h4/finite__map/f__o__f f g) x = h4/finite__map/FAPPLY f (h4/finite__map/FAPPLY g x)
% Assm [h4s_alists_FDOMu_u_alistu_u_tou_u_fmap]: !al. h4/finite__map/FDOM (h4/alist/alist__to__fmap al) = h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST al)
% Assm [h4s_lists_FINITEu_u_LISTu_u_TOu_u_SET]: !l. h4/pred__set/FINITE (h4/list/LIST__TO__SET l)
% Assm [h4s_alists_ALOOKUPu_u_SOMEu_u_FAPPLYu_u_alistu_u_tou_u_fmap]: !v k al. h4/alist/ALOOKUP al k = h4/option/SOME v ==> h4/finite__map/FAPPLY (h4/alist/alist__to__fmap al) k = v
% Assm [h4s_alists_ALOOKUPu_u_FAILS]: !x l. h4/alist/ALOOKUP l x = h4/option/NONE <=> (!k v. h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET l) ==> ~(k = x))
% Assm [h4s_pairs_pairu_u_CASES]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (h4/option/SOME x) = x
% Assm [h4s_predu_u_sets_INJu_u_DEF]: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> happ f x = happ f y ==> x = y)
% 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_INJu_u_INSERT]: !x t s f. h4/pred__set/INJ f (h4/pred__set/INSERT x s) t <=> h4/pred__set/INJ f s t /\ h4/bool/IN (happ f x) t /\ (!y. h4/bool/IN y s /\ happ f x = happ f y ==> x = y)
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_predu_u_sets_INTERu_u_EMPTYu_c0]: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_bools_CONDu_u_EXPAND]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% Assm [h4s_bools_CONDu_u_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% Assm [h4s_bools_CONDu_u_RATOR]: !x g f b. happ (h4/bool/COND b f g) x = h4/bool/COND b (happ f x) (happ g x)
% Assm [h4s_lists_ALLu_u_DISTINCT0u_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 [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_lists_SETu_u_TOu_u_LISTu_u_primitive]: !_2. (!SET__TO__LIST s. ?v. (v <=> s = h4/pred__set/EMPTY) /\ happ (happ _2 SET__TO__LIST) s = h4/combin/I (h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND v h4/list/NIL (h4/list/CONS (h4/pred__set/CHOICE s) (happ SET__TO__LIST (h4/pred__set/REST s)))) h4/bool/ARB)) ==> (!_1. (!SET__TO__LIST. happ _1 SET__TO__LIST = happ _2 SET__TO__LIST) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!s. h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> happ (happ R (h4/pred__set/REST s)) s)) ==> h4/list/SET__TO__LIST = h4/relation/WFREC (h4/min/_40 _0) _1))
% Goal: !fm2 fm1. h4/finite__map/FDOM fm1 = h4/finite__map/FDOM fm2 ==> h4/list/MAP h4/pair/FST (h4/alist/fmap__to__alist fm1) = h4/list/MAP h4/pair/FST (h4/alist/fmap__to__alist fm2)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1364512,TV_Q1364508]: ![V_f, V_g]: (![V_x]: s(TV_Q1364508,happ(s(t_fun(TV_Q1364512,TV_Q1364508),V_f),s(TV_Q1364512,V_x))) = s(TV_Q1364508,happ(s(t_fun(TV_Q1364512,TV_Q1364508),V_g),s(TV_Q1364512,V_x))) => s(t_fun(TV_Q1364512,TV_Q1364508),V_f) = s(t_fun(TV_Q1364512,TV_Q1364508),V_g))).
fof(ah4s_finiteu_u_maps_MAPu_u_KEYSu_u_witness, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_2]: (![V_f, V_fm]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_uu_2),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_bools_cond(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(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_27c),V_fm))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27b,TV_u_27a),h4s_predu_u_sets_linv(s(t_fun(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_27c),V_fm))))),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_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))))))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27b,TV_u_27c),h4s_bools_arb),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_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))))))) => ![V_uu_1]: (![V_f]: s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_uu_2),s(t_fun(TV_u_27a,TV_u_27b),V_f))) => ![V_uu_0]: (![V_m]: (p(s(t_bool,happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),t_bool),V_uu_0),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_m)))) <=> ![V_f, V_fm]: (s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_m),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))) = 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_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))) & (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(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_27c),V_fm))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => ![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_27c),V_fm)))))) => s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_m),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),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_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(TV_u_27a,V_x))))))) => p(s(t_bool,h4s_bools_let(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),t_bool),V_uu_0),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c))),V_uu_1)))))))).
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_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_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_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(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_x)))).
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_fu_u_ou_u_fu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_g, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_bools_in(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_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))) => ![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_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(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_predu_u_sets_inter(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_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))))).
fof(ah4s_finiteu_u_maps_FMAPu_u_MAP2u_u_def, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_f, V_m, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),TV_u_27b),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_m))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_m),s(TV_u_27a,V_x))))))) => ![V_m, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fmapu_u_map2(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_m))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),TV_u_27b),V_f))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_m))),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),V_m))))))).
fof(ah4s_finiteu_u_maps_MAPu_u_KEYSu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_fm, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))) = 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_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm)))))).
fof(ah4s_finiteu_u_maps_fu_u_ou_u_DEF, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_g, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))) => ![V_g, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fu_u_o(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_finiteu_u_maps_MAPu_u_KEYSu_u_FUPDATE, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_v, V_k, V_fm, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(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_k),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),V_fm))))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27c,V_v))))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_k))),s(TV_u_27c,V_v))))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_fu_u_o, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_g, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))) => ![V_g, V_f]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),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_fu_u_o(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f)))))))).
fof(ah4s_finiteu_u_maps_FMAPu_u_MAP2u_u_THMu_c0, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_m, 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_fmapu_u_map2(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_m))))) = 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),V_m)))).
fof(ah4s_finiteu_u_maps_FINITEu_u_PREDu_u_11, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_g, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))) => ![V_g]: (![V_x, V_y]: (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => ![V_f]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))))))).
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_finiteu_u_maps_MAPu_u_KEYSu_u_usingu_u_LINV, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_fm, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(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_27c),V_fm))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27b,TV_u_27a),h4s_predu_u_sets_linv(s(t_fun(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_27c),V_fm))))),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_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))))))))))).
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_alists_alistu_u_tou_u_fmapu_u_tou_u_alist, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_al, V_k]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,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_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_uu_0),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al))),s(TV_u_27a,V_k))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27b,h4s_options_the(s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al),s(TV_u_27a,V_k))))))) => ![V_al]: s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_alists_fmapu_u_tou_u_alist(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al))))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,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_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_uu_0),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),h4s_lists_setu_u_tou_u_list),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_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al))))))))))).
fof(ah4s_alists_ALLu_u_DISTINCTu_u_fmapu_u_tou_u_alistu_u_keys, axiom, ![TV_u_27a,TV_u_27b]: ![V_fm]: p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_alists_fmapu_u_tou_u_alist(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm))))))))).
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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_finiteu_u_maps_FUNu_u_FMAPu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_P)))) => (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_funu_u_fmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_P))))) = s(t_fun(TV_u_27a,t_bool),V_P) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => 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_funu_u_fmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_FINITE, axiom, ![TV_u_27a,TV_u_27b]: ![V_fm]: p(s(t_bool,h4s_predu_u_sets_finite(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_fm))))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_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_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_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
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_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_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_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_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_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_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_finiteu_u_maps_MAPu_u_KEYSu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_fm, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(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_27c),V_fm))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))) => ![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_27c),V_fm)))))) => s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),h4s_finiteu_u_maps_mapu_u_keys(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),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_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(TV_u_27a,V_x)))))).
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_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,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
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_predu_u_sets_INu_u_INTER, 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_inter(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_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_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_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,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (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_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
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,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_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_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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_predu_u_sets_GSPECu_u_OR, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_P, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_P, V_Q, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (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)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_Q))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_Q, V_P]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_Q))))) = 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_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_Q)))))))))).
fof(ah4s_predu_u_sets_GSPECu_u_EQ, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_y, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_y]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(TV_u_27a,V_y))))) = 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),h4s_predu_u_sets_empty))))).
fof(ah4s_predu_u_sets_GSPECu_u_F, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,f))) => s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
fof(ah4s_predu_u_sets_FINITEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(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_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),V_t))))))).
fof(ah4s_predu_u_sets_SINGu_u_FINITE, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_sing(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),V_s)))))).
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_SING0, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_predu_u_sets_sing(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),h4s_predu_u_sets_empty))))))).
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_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_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
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_lists_MAPu_u_EQu_u_f, axiom, ![TV_u_27b,TV_u_27a]: ![V_l, V_f2, V_f1]: (s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f2),s(t_h4s_lists_list(TV_u_27a),V_l))) <=> ![V_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_e))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f2),s(TV_u_27a,V_e)))))).
fof(ah4s_alists_fmapu_u_tou_u_alistu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_s, V_k]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_s))),s(TV_u_27a,V_k))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_s),s(TV_u_27a,V_k))))) => ![V_s]: s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_alists_fmapu_u_tou_u_alist(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_s))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_s))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),h4s_lists_setu_u_tou_u_list),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_s))))))))).
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_pairs_CLOSEDu_u_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_lists_MEMu_u_SETu_u_TOu_u_LIST, 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_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),happ(s(t_fun(t_fun(TV_u_27a,t_bool),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_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
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,f))).
fof(ah4s_predu_u_sets_IMAGEu_u_FINITE, 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]: p(s(t_bool,h4s_predu_u_sets_finite(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_IMAGEu_u_IN, axiom, ![TV_u_27b,TV_u_27a]: ![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),V_s)))) => ![V_f]: p(s(t_bool,h4s_bools_in(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_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![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)))) => 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_predu_u_sets_LINVu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) => ![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)))) => s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),h4s_predu_u_sets_linv(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s))),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_27a,V_x)))).
fof(ah4s_predu_u_sets_SUBSETu_u_INTERu_u_ABSORPTION, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(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),h4s_predu_u_sets_inter(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_s))).
fof(ah4s_finiteu_u_maps_fu_u_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_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(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_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(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,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(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_alists_FDOMu_u_alistu_u_tou_u_fmap, axiom, ![TV_u_27a,TV_u_27b]: ![V_al]: 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_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al))))) = 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_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al)))))).
fof(ah4s_lists_FINITEu_u_LISTu_u_TOu_u_SET, axiom, ![TV_u_27a]: ![V_l]: p(s(t_bool,h4s_predu_u_sets_finite(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_alists_ALOOKUPu_u_SOMEu_u_FAPPLYu_u_alistu_u_tou_u_fmap, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_k, V_al]: (s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al),s(TV_u_27a,V_k))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_v))) => s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_alists_alistu_u_tou_u_fmap(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_al))),s(TV_u_27a,V_k))) = s(TV_u_27b,V_v))).
fof(ah4s_alists_ALOOKUPu_u_FAILS, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_l]: (s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_l),s(TV_u_27b,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> ![V_k, V_v]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_k),s(TV_u_27a,V_v))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_l)))))) => ~ (s(TV_u_27b,V_k) = s(TV_u_27b,V_x))))).
fof(ah4s_pairs_pairu_u_CASES, 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_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_THEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_predu_u_sets_INJu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![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)))) => p(s(t_bool,h4s_bools_in(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),V_t))))) & ![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),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s))))) => (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)))))).
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_INJu_u_INSERT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(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),V_t)))) <=> (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & (p(s(t_bool,h4s_bools_in(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),V_t)))) & ![V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) & 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)))))).
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_predu_u_sets_INu_u_UNIV, 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_univ))))).
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_predu_u_sets_INTERu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_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_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_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_EXPAND, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((~ (p(s(t_bool,V_b))) | p(s(t_bool,V_t1))) & (p(s(t_bool,V_b)) | p(s(t_bool,V_t2)))))).
fof(ah4s_bools_CONDu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),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)))))).
fof(ah4s_bools_CONDu_u_RATOR, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_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),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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_SETu_u_TOu_u_LISTu_u_primitive, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_SETu_u_TOu_u_LIST, 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),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a))),V_uu_2),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),V_SETu_u_TOu_u_LIST))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_lists_list(TV_u_27a),h4s_combins_i(s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_predu_u_sets_finite(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),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),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),V_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))))))))),s(t_h4s_lists_list(TV_u_27a),h4s_bools_arb)))))) => ![V_uu_1]: (![V_SETu_u_TOu_u_LIST]: s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a))),V_uu_1),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),V_SETu_u_TOu_u_LIST))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a))),V_uu_2),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),V_SETu_u_TOu_u_LIST))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),t_bool),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_R)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),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,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_R),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))))))) => s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),h4s_lists_setu_u_tou_u_list) = s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a)),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_lists_list(TV_u_27a))),V_uu_1))))))).
fof(ch4s_alists_fmapu_u_tou_u_alistu_u_preservesu_u_FDOM, conjecture, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_fm2, V_fm1]: (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_fm1))) = 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),V_fm2))) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_alists_fmapu_u_tou_u_alist(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm1))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_alists_fmapu_u_tou_u_alist(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm2))))))).
