%   ORIGINAL: h4/fmaptree/relrec__cases
% 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/fmaptree/relrec__def: h4/fmaptree/relrec = (\h a0 a1. !relrec_27. (!a00 a10. (?i fm rfm. a00 = h4/fmaptree/FTNode i fm /\ a10 = h i rfm fm /\ h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> relrec_27 (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d))) ==> relrec_27 a00 a10) ==> relrec_27 a0 a1)
% 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/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/list/list__Axiom: !f1 f0. ?fn. fn h4/list/NIL = f0 /\ (!a0 a1. fn (h4/list/CONS a0 a1) = f1 a0 a1 (fn a1))
% Assm: h4/bool/TRUTH: T
% 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/finite__map/fmap__rel__def: !f2 f1 R. h4/finite__map/fmap__rel R f1 f2 <=> h4/finite__map/FDOM f2 = h4/finite__map/FDOM f1 /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f1) ==> R (h4/finite__map/FAPPLY f1 x) (h4/finite__map/FAPPLY f2 x))
% 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__MAP2__THM_c1: !x m f. h4/bool/IN x (h4/finite__map/FDOM m) ==> h4/finite__map/FAPPLY (h4/finite__map/FMAP__MAP2 f m) x = f (h4/pair/_2C x (h4/finite__map/FAPPLY m x))
% 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__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> 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/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/fmaptree/relrec__ind: !relrec_27 h. (!i fm rfm. h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> relrec_27 (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d)) ==> relrec_27 (h4/fmaptree/FTNode i fm) (h i rfm fm)) ==> (!a0 a1. h4/fmaptree/relrec h a0 a1 ==> relrec_27 a0 a1)
% Assm: h4/fmaptree/relrec__strongind: !relrec_27 h. (!i fm rfm. h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> h4/fmaptree/relrec h (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d) /\ relrec_27 (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d)) ==> relrec_27 (h4/fmaptree/FTNode i fm) (h i rfm fm)) ==> (!a0 a1. h4/fmaptree/relrec h a0 a1 ==> relrec_27 a0 a1)
% Assm: h4/fmaptree/relrec__rules: !rfm i h fm. h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> h4/fmaptree/relrec h (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d)) ==> h4/fmaptree/relrec h (h4/fmaptree/FTNode i fm) (h i rfm fm)
% 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/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__ALL: !Q P. (!x. P x ==> Q x) ==> (!x. P x) ==> (!x. Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__IMP: !z y x w. (y ==> x) /\ (z ==> w) ==> (x ==> z) ==> y ==> w
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/finite__map/fmap__rel__OPTREL__FLOOKUP: !f2 f1 R. h4/finite__map/fmap__rel R f1 f2 <=> (!k. h4/option/OPTREL R (h4/finite__map/FLOOKUP f1 k) (h4/finite__map/FLOOKUP f2 k))
% Assm: h4/finite__map/o__f__DOMSUB: !k g fm. h4/finite__map/fdomsub (h4/finite__map/o__f g fm) k = h4/finite__map/o__f g (h4/finite__map/fdomsub fm k)
% Assm: h4/fmaptree/update__at__path__def_c1: !t h ft a. h4/fmaptree/update__at__path (h4/list/CONS h t) a ft = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM (h4/fmaptree/map ft))) (h4/option/option__CASE (h4/fmaptree/update__at__path t a (h4/finite__map/FAPPLY (h4/fmaptree/map ft) h)) h4/option/NONE (\ft_27. h4/option/SOME (h4/fmaptree/FTNode (h4/fmaptree/item ft) (h4/finite__map/FUPDATE (h4/fmaptree/map ft) (h4/pair/_2C h ft_27))))) h4/option/NONE
% Assm: h4/fmaptree/ft__ind: !P. (!a fm. (!k. h4/bool/IN k (h4/finite__map/FDOM fm) ==> P (h4/finite__map/FAPPLY fm k)) ==> P (h4/fmaptree/FTNode a fm)) ==> (!ft. P ft)
% 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/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/finite__map/FDOM__FINITE: !fm. h4/pred__set/FINITE (h4/finite__map/FDOM fm)
% Assm: h4/fmaptree/fupd__at__path__def_c1: !t h ft f. h4/fmaptree/fupd__at__path (h4/list/CONS h t) f ft = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM (h4/fmaptree/map ft))) (h4/option/option__CASE (h4/fmaptree/fupd__at__path t f (h4/finite__map/FAPPLY (h4/fmaptree/map ft) h)) h4/option/NONE (\ft_27. h4/option/SOME (h4/fmaptree/FTNode (h4/fmaptree/item ft) (h4/finite__map/FUPDATE (h4/fmaptree/map ft) (h4/pair/_2C h ft_27))))) h4/option/NONE
% Assm: h4/fmaptree/fmaptree__nchotomy: !ft. ?i fm. ft = h4/fmaptree/FTNode i fm
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% 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_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% 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/NOT__IN__EMPTY: !x. ~h4/bool/IN x 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/OR__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/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% 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/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x 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__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% 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/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% 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/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/finite__map/DOMSUB__FAPPLY__THM: !k2 k1 fm. h4/finite__map/FAPPLY (h4/finite__map/fdomsub fm k1) k2 = h4/bool/COND (k1 = k2) (h4/finite__map/FAPPLY h4/finite__map/FEMPTY k2) (h4/finite__map/FAPPLY fm k2)
% Assm: h4/finite__map/FDOM__DOMSUB: !k fm. h4/finite__map/FDOM (h4/finite__map/fdomsub fm k) = h4/pred__set/DELETE (h4/finite__map/FDOM fm) k
% Assm: h4/pred__set/IN__DELETE: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/fmaptree/fmap__bij__thm_c1: !r. h4/fmaptree/wf r <=> h4/fmaptree/toF (h4/fmaptree/fromF r) = r
% Assm: h4/fmaptree/fmap__bij__thm_c0: !a. h4/fmaptree/fromF (h4/fmaptree/toF a) = a
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/pred__set/INTER__EMPTY_c0: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% 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/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% 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/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% 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/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/bool/COND__EXPAND: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% 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/finite__map/FLOOKUP__DEF: !x f. h4/finite__map/FLOOKUP f x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/option/SOME (h4/finite__map/FAPPLY f x)) h4/option/NONE
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/option/IF__EQUALS__OPTION_c2: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/option/OPTREL__def: !y x R. h4/option/OPTREL R x y <=> x = h4/option/NONE /\ y = h4/option/NONE \/ (?x0 y0. x = h4/option/SOME x0 /\ y = h4/option/SOME y0 /\ R x0 y0)
% Assm: h4/option/IF__EQUALS__OPTION_c0: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/fmaptree/wf__def: h4/fmaptree/wf = (\a0. !wf_27. (!a00. (?a fm. a00 = h4/fmaptree/construct a fm /\ (!k. h4/bool/IN k (h4/finite__map/FDOM fm) ==> wf_27 (h4/finite__map/FAPPLY fm k))) ==> wf_27 a00) ==> wf_27 a0)
% Assm: h4/bool/EQ__REFL: !x. x = x
% 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/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/fmaptree/FTNode__def: !i fm. h4/fmaptree/FTNode i fm = h4/fmaptree/fromF (h4/fmaptree/construct i (h4/finite__map/o__f h4/fmaptree/toF fm))
% Assm: h4/fmaptree/construct__def: !kl kfm a. h4/fmaptree/construct a kfm kl = h4/list/list__CASE kl (h4/option/SOME a) (\h t. h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM kfm)) (h4/finite__map/FAPPLY kfm h t) h4/option/NONE)
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/BOOL__EQ__DISTINCT_c0: ~(T <=> F)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Goal: !h a1 a0. h4/fmaptree/relrec h a0 a1 <=> (?i fm rfm. a0 = h4/fmaptree/FTNode i fm /\ a1 = h i rfm fm /\ h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> h4/fmaptree/relrec h (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d)))
%   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_fmaptrees_relrecu_u_def]: !x x x. h4/fmaptree/relrec x x x <=> (!relrec_27. (!a00 a10. (?i fm rfm. a00 = h4/fmaptree/FTNode i fm /\ a10 = happ (happ (happ x i) rfm) fm /\ h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> happ (happ relrec_27 (h4/finite__map/FAPPLY fm d)) (h4/finite__map/FAPPLY rfm d))) ==> happ (happ relrec_27 a00) a10) ==> happ (happ relrec_27 x) x)
% 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_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_lists_listu_u_Axiom]: !f1 f0. ?fn. happ fn h4/list/NIL = f0 /\ (!a0 a1. happ fn (h4/list/CONS a0 a1) = happ (happ (happ f1 a0) a1) (happ fn a1))
% Assm [h4s_bools_TRUTH]: T
% 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_finiteu_u_maps_fmapu_u_relu_u_def]: !f2 f1 R. h4/finite__map/fmap__rel R f1 f2 <=> h4/finite__map/FDOM f2 = h4/finite__map/FDOM f1 /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f1) ==> happ (happ R (h4/finite__map/FAPPLY f1 x)) (h4/finite__map/FAPPLY f2 x))
% 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_MAP2u_u_THMu_c1]: !x m f. h4/bool/IN x (h4/finite__map/FDOM m) ==> h4/finite__map/FAPPLY (h4/finite__map/FMAP__MAP2 f m) x = happ f (h4/pair/_2C x (h4/finite__map/FAPPLY m x))
% 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_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> 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_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_fmaptrees_relrecu_u_ind]: !relrec_27 h. (!i fm rfm. h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> happ (happ relrec_27 (h4/finite__map/FAPPLY fm d)) (h4/finite__map/FAPPLY rfm d)) ==> happ (happ relrec_27 (h4/fmaptree/FTNode i fm)) (happ (happ (happ h i) rfm) fm)) ==> (!a0 a1. h4/fmaptree/relrec h a0 a1 ==> happ (happ relrec_27 a0) a1)
% Assm [h4s_fmaptrees_relrecu_u_strongind]: !relrec_27 h. (!i fm rfm. h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> h4/fmaptree/relrec h (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d) /\ happ (happ relrec_27 (h4/finite__map/FAPPLY fm d)) (h4/finite__map/FAPPLY rfm d)) ==> happ (happ relrec_27 (h4/fmaptree/FTNode i fm)) (happ (happ (happ h i) rfm) fm)) ==> (!a0 a1. h4/fmaptree/relrec h a0 a1 ==> happ (happ relrec_27 a0) a1)
% Assm [h4s_fmaptrees_relrecu_u_rules]: !rfm i h fm. h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> h4/fmaptree/relrec h (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d)) ==> h4/fmaptree/relrec h (h4/fmaptree/FTNode i fm) (happ (happ (happ h i) rfm) fm)
% 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_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_ALL]: !Q P. (!x. happ P x ==> happ Q x) ==> (!x. happ P x) ==> (!x. happ Q x)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_IMP]: !z y x w. (y ==> x) /\ (z ==> w) ==> (x ==> z) ==> y ==> w
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> 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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_finiteu_u_maps_fmapu_u_relu_u_OPTRELu_u_FLOOKUP]: !f2 f1 R. h4/finite__map/fmap__rel R f1 f2 <=> (!k. h4/option/OPTREL R (h4/finite__map/FLOOKUP f1 k) (h4/finite__map/FLOOKUP f2 k))
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_DOMSUB]: !k g fm. h4/finite__map/fdomsub (h4/finite__map/o__f g fm) k = h4/finite__map/o__f g (h4/finite__map/fdomsub fm k)
% Assm [h4s_fmaptrees_updateu_u_atu_u_pathu_u_defu_c1]: !_0. (!ft h ft_27. happ (happ (happ _0 ft) h) ft_27 = h4/option/SOME (h4/fmaptree/FTNode (h4/fmaptree/item ft) (h4/finite__map/FUPDATE (h4/fmaptree/map ft) (h4/pair/_2C h ft_27)))) ==> (!t h ft a. h4/fmaptree/update__at__path (h4/list/CONS h t) a ft = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM (h4/fmaptree/map ft))) (h4/option/option__CASE (h4/fmaptree/update__at__path t a (h4/finite__map/FAPPLY (h4/fmaptree/map ft) h)) h4/option/NONE (happ (happ _0 ft) h)) h4/option/NONE)
% Assm [h4s_fmaptrees_ftu_u_ind]: !P. (!a fm. (!k. h4/bool/IN k (h4/finite__map/FDOM fm) ==> happ P (h4/finite__map/FAPPLY fm k)) ==> happ P (h4/fmaptree/FTNode a fm)) ==> (!ft. happ P ft)
% 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_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_finiteu_u_maps_FDOMu_u_FINITE]: !fm. h4/pred__set/FINITE (h4/finite__map/FDOM fm)
% Assm [h4s_fmaptrees_fupdu_u_atu_u_pathu_u_defu_c1]: !_0. (!ft h ft_27. happ (happ (happ _0 ft) h) ft_27 = h4/option/SOME (h4/fmaptree/FTNode (h4/fmaptree/item ft) (h4/finite__map/FUPDATE (h4/fmaptree/map ft) (h4/pair/_2C h ft_27)))) ==> (!t h ft f. h4/fmaptree/fupd__at__path (h4/list/CONS h t) f ft = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM (h4/fmaptree/map ft))) (h4/option/option__CASE (h4/fmaptree/fupd__at__path t f (h4/finite__map/FAPPLY (h4/fmaptree/map ft) h)) h4/option/NONE (happ (happ _0 ft) h)) h4/option/NONE)
% Assm [h4s_fmaptrees_fmaptreeu_u_nchotomy]: !ft. ?i fm. ft = h4/fmaptree/FTNode i fm
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% 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_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% 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_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x 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_ORu_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_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% 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_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x 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_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% 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_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% 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_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_finiteu_u_maps_DOMSUBu_u_FAPPLYu_u_THM]: !k2 k1 fm. ?v. (v <=> k1 = k2) /\ h4/finite__map/FAPPLY (h4/finite__map/fdomsub fm k1) k2 = h4/bool/COND v (h4/finite__map/FAPPLY h4/finite__map/FEMPTY k2) (h4/finite__map/FAPPLY fm k2)
% Assm [h4s_finiteu_u_maps_FDOMu_u_DOMSUB]: !k fm. h4/finite__map/FDOM (h4/finite__map/fdomsub fm k) = h4/pred__set/DELETE (h4/finite__map/FDOM fm) k
% Assm [h4s_predu_u_sets_INu_u_DELETE]: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_fmaptrees_fmapu_u_biju_u_thmu_c1]: !r. h4/fmaptree/wf r <=> happ h4/fmaptree/toF (h4/fmaptree/fromF r) = r
% Assm [h4s_fmaptrees_fmapu_u_biju_u_thmu_c0]: !a. h4/fmaptree/fromF (happ h4/fmaptree/toF a) = a
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~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_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_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% 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_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% 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_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_bools_CONDu_u_EXPAND]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% 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_finiteu_u_maps_FLOOKUPu_u_DEF]: !x f. h4/finite__map/FLOOKUP f x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/option/SOME (h4/finite__map/FAPPLY f x)) h4/option/NONE
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c2]: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_options_OPTRELu_u_def]: !y x R. h4/option/OPTREL R x y <=> x = h4/option/NONE /\ y = h4/option/NONE \/ (?x0 y0. x = h4/option/SOME x0 /\ y = h4/option/SOME y0 /\ happ (happ R x0) y0)
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c0]: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_fmaptrees_wfu_u_def]: !x. h4/fmaptree/wf x <=> (!wf_27. (!a00. (?a fm. a00 = h4/fmaptree/construct a fm /\ (!k. h4/bool/IN k (h4/finite__map/FDOM fm) ==> happ wf_27 (h4/finite__map/FAPPLY fm k))) ==> happ wf_27 a00) ==> happ wf_27 x)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% 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_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_combins_ou_u_THM]: !x g f. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_fmaptrees_FTNodeu_u_def]: !i fm. h4/fmaptree/FTNode i fm = h4/fmaptree/fromF (h4/fmaptree/construct i (h4/finite__map/o__f h4/fmaptree/toF fm))
% Assm [h4s_fmaptrees_constructu_u_def]: !_1. (!kfm h t. happ (happ (happ _1 kfm) h) t = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM kfm)) (happ (h4/finite__map/FAPPLY kfm h) t) h4/option/NONE) ==> (!_0. (!kfm h. happ (happ _0 kfm) h = happ (happ _1 kfm) h) ==> (!kl kfm a. happ (h4/fmaptree/construct a kfm) kl = h4/list/list__CASE kl (h4/option/SOME a) (happ _0 kfm)))
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0]: ~(T <=> F)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Goal: !h a1 a0. h4/fmaptree/relrec h a0 a1 <=> (?i fm rfm. a0 = h4/fmaptree/FTNode i fm /\ a1 = happ (happ (happ h i) rfm) fm /\ h4/finite__map/FDOM rfm = h4/finite__map/FDOM fm /\ (!d. h4/bool/IN d (h4/finite__map/FDOM fm) ==> h4/fmaptree/relrec h (h4/finite__map/FAPPLY fm d) (h4/finite__map/FAPPLY rfm d)))
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_Q1348603,TV_Q1348599]: ![V_f, V_g]: (![V_x]: s(TV_Q1348599,happ(s(t_fun(TV_Q1348603,TV_Q1348599),V_f),s(TV_Q1348603,V_x))) = s(TV_Q1348599,happ(s(t_fun(TV_Q1348603,TV_Q1348599),V_g),s(TV_Q1348603,V_x))) => s(t_fun(TV_Q1348603,TV_Q1348599),V_f) = s(t_fun(TV_Q1348603,TV_Q1348599),V_g))).
fof(ah4s_fmaptrees_relrecu_u_def, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_x0, V_x1]: (p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_x),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_x0),s(TV_u_27c,V_x1)))) <=> ![V_relrecu_27]: (![V_a00, V_a10]: (?[V_i, V_fm, V_rfm]: (s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a00) = s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_fmaptrees_ftnode(s(TV_u_27a,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & (s(TV_u_27c,V_a10) = s(TV_u_27c,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_x),s(TV_u_27a,V_i))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),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),V_rfm))) = s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & ![V_d]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_d),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm),s(TV_u_27b,V_d))))),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm),s(TV_u_27b,V_d)))))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a00))),s(TV_u_27c,V_a10))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_x0))),s(TV_u_27c,V_x1))))))).
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_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_lists_listu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f1, V_f0]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_f0) & ![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),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(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b))),V_f1),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),V_a1))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_finiteu_u_maps_fmapu_u_relu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![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_fun(TV_u_27c,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2))) = s(t_fun(TV_u_27c,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27c,V_x),s(t_fun(TV_u_27c,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1)))))) => 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,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1),s(TV_u_27c,V_x))))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2),s(TV_u_27c,V_x)))))))))).
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_MAP2u_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_m, 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),V_m)))))) => 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_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(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))))))))).
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_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_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_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_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_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
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_fmaptrees_relrecu_u_ind, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_relrecu_27, V_h]: (![V_i, V_fm, V_rfm]: ((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_rfm))) = s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & ![V_d]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_d),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm),s(TV_u_27b,V_d))))),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm),s(TV_u_27b,V_d)))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_fmaptrees_ftnode(s(TV_u_27a,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))))),s(TV_u_27c,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(TV_u_27a,V_i))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))))))) => ![V_a0, V_a1]: (p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a0),s(TV_u_27c,V_a1)))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a0))),s(TV_u_27c,V_a1))))))).
fof(ah4s_fmaptrees_relrecu_u_strongind, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_relrecu_27, V_h]: (![V_i, V_fm, V_rfm]: ((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_rfm))) = s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & ![V_d]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_d),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)))))) => (p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm),s(TV_u_27b,V_d))),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm),s(TV_u_27b,V_d)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm),s(TV_u_27b,V_d))))),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm),s(TV_u_27b,V_d))))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_fmaptrees_ftnode(s(TV_u_27a,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))))),s(TV_u_27c,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(TV_u_27a,V_i))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))))))) => ![V_a0, V_a1]: (p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a0),s(TV_u_27c,V_a1)))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),t_fun(TV_u_27c,t_bool)),V_relrecu_27),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a0))),s(TV_u_27c,V_a1))))))).
fof(ah4s_fmaptrees_relrecu_u_rules, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_rfm, V_i, V_h, 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),V_rfm))) = s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & ![V_d]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_d),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)))))) => p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm),s(TV_u_27b,V_d))),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm),s(TV_u_27b,V_d)))))))) => p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_fmaptrees_ftnode(s(TV_u_27a,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))),s(TV_u_27c,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(TV_u_27a,V_i))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)))))))).
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_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_MONOu_u_ALL, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_IMP, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_y)) => p(s(t_bool,V_x))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) => p(s(t_bool,V_w)))))).
fof(ah4s_bools_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_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_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_finiteu_u_maps_fmapu_u_relu_u_OPTRELu_u_FLOOKUP, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![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)))) <=> ![V_k]: p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1),s(TV_u_27c,V_k))),s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2),s(TV_u_27c,V_k)))))))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_DOMSUB, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_k, V_g, V_fm]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fdomsub(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_g),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm))),s(TV_u_27a,V_k))) = 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_g),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),V_fm),s(TV_u_27a,V_k)))))).
fof(ah4s_fmaptrees_updateu_u_atu_u_pathu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_ft, V_h, V_ftu_27]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))))),V_uu_0),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(TV_u_27a,V_h))),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ftu_27))) = s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_some(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,h4s_fmaptrees_item(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_pairs_u_2c(s(TV_u_27a,V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ftu_27))))))))) => ![V_t, V_h, V_ft, V_a]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_updateu_u_atu_u_path(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(TV_u_27b,V_a),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))) = s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_h),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))))))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_updateu_u_atu_u_path(s(t_h4s_lists_list(TV_u_27a),V_t),s(TV_u_27b,V_a),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(TV_u_27a,V_h))))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_none),s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))))),V_uu_0),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(TV_u_27a,V_h))))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_none))))).
fof(ah4s_fmaptrees_ftu_u_ind, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_a, V_fm]: (![V_k]: (p(s(t_bool,h4s_bools_in(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,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm),s(TV_u_27a,V_k))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,V_a),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm))))))) => ![V_ft]: p(s(t_bool,happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft)))))).
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_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_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_fmaptrees_fupdu_u_atu_u_pathu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_ft, V_h, V_ftu_27]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))))),V_uu_0),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(TV_u_27a,V_h))),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ftu_27))) = s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_some(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,h4s_fmaptrees_item(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_pairs_u_2c(s(TV_u_27a,V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ftu_27))))))))) => ![V_t, V_h, V_ft, V_f]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_fupdu_u_atu_u_path(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_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))),V_f),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))) = s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_h),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))))))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_fupdu_u_atu_u_path(s(t_h4s_lists_list(TV_u_27a),V_t),s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))),V_f),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(TV_u_27a,V_h))))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_none),s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b))))),V_uu_0),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(TV_u_27a,V_h))))),s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_options_none))))).
fof(ah4s_fmaptrees_fmaptreeu_u_nchotomy, axiom, ![TV_u_27a,TV_u_27b]: ![V_ft]: ?[V_i, V_fm]: s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft) = s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_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_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_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,f),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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_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,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_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_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_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_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_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_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_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_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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
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_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> 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_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_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_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_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_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_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_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_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_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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_finiteu_u_maps_DOMSUBu_u_FAPPLYu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_k2, V_k1, V_fm]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_k1) = s(TV_u_27a,V_k2)) & 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_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k1))),s(TV_u_27a,V_k2))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,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_finiteu_u_maps_fempty),s(TV_u_27a,V_k2))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k2))))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_DOMSUB, axiom, ![TV_u_27b,TV_u_27a]: ![V_k, V_fm]: 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_fdomsub(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(TV_u_27a,V_k))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(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))),s(TV_u_27a,V_k)))).
fof(ah4s_predu_u_sets_INu_u_DELETE, 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_delete(s(t_fun(TV_u_27a,t_bool),V_s),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)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & 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,V_a)))))).
fof(ah4s_fmaptrees_fmapu_u_biju_u_thmu_c1, axiom, ![TV_u_27key,TV_u_27value]: ![V_r]: (p(s(t_bool,h4s_fmaptrees_wf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r)))) <=> s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value))),h4s_fmaptrees_tof),s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),h4s_fmaptrees_fromf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r))))) = s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),V_r))).
fof(ah4s_fmaptrees_fmapu_u_biju_u_thmu_c0, axiom, ![TV_u_27key,TV_u_27value]: ![V_a]: s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),h4s_fmaptrees_fromf(s(t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),t_fun(t_h4s_lists_list(TV_u_27key),t_h4s_options_option(TV_u_27value))),h4s_fmaptrees_tof),s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),V_a))))) = s(t_h4s_fmaptrees_fmaptree(TV_u_27key,TV_u_27value),V_a)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_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_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_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_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_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_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_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_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_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_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_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_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_finiteu_u_maps_FLOOKUPu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(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(t_h4s_options_option(TV_u_27b),h4s_options_some(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(t_h4s_options_option(TV_u_27b),h4s_options_none)))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),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_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> (p(s(t_bool,V_P)) & s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_bools_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_options_OPTRELu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27b),V_y)))) <=> ((s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) & s(t_h4s_options_option(TV_u_27b),V_y) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)) | ?[V_x0, V_y0]: (s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x0))) & (s(t_h4s_options_option(TV_u_27b),V_y) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_y0))) & 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_x0))),s(TV_u_27b,V_y0))))))))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c0, axiom, ![TV_u_27a]: ![V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),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_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> ~ (p(s(t_bool,V_P))))).
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_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) & ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_fmaptrees_wfu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_fmaptrees_wf(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),V_x)))) <=> ![V_wfu_27]: (![V_a00]: (?[V_a, V_fm]: (s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),V_a00) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_fmaptrees_construct(s(TV_u_27a,V_a),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_fm))) & ![V_k]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_k),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_fm)))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),t_bool),V_wfu_27),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_fm),s(TV_u_27b,V_k)))))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),t_bool),V_wfu_27),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),t_bool),V_wfu_27),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),V_x))))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_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_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_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_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_fmaptrees_FTNodeu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_fm]: s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm))) = s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_fromf(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_fmaptrees_construct(s(TV_u_27b,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(TV_u_27b))),h4s_finiteu_u_maps_ou_u_f(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(TV_u_27b))),h4s_fmaptrees_tof),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_fm)))))))).
fof(ah4s_fmaptrees_constructu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_1]: (![V_kfm, V_h, V_t]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)))),V_uu_1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27b,V_h),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))))),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => ![V_uu_0]: (![V_kfm, V_h]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))),s(TV_u_27b,V_h))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)))),V_uu_1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))),s(TV_u_27b,V_h))) => ![V_kl, V_kfm, V_a]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_fmaptrees_construct(s(TV_u_27a,V_a),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm))),s(t_h4s_lists_list(TV_u_27b),V_kl))) = s(t_h4s_options_option(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27b),V_kl),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a)))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_options_option(TV_u_27a))),V_kfm)))))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: 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_bools_BOOLu_u_EQu_u_DISTINCTu_c0, axiom, ~ (s(t_bool,t) = s(t_bool,f))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(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(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = 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(ch4s_fmaptrees_relrecu_u_cases, conjecture, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_h, V_a1, V_a0]: (p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a0),s(TV_u_27c,V_a1)))) <=> ?[V_i, V_fm, V_rfm]: (s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),V_a0) = s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_fmaptrees_ftnode(s(TV_u_27a,V_i),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & (s(TV_u_27c,V_a1) = s(TV_u_27c,happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(TV_u_27a,V_i))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),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),V_rfm))) = s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm))) & ![V_d]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_d),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)))))) => p(s(t_bool,h4s_fmaptrees_relrec(s(t_fun(TV_u_27a,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),TV_u_27c))),V_h),s(t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm),s(TV_u_27b,V_d))),s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_rfm),s(TV_u_27b,V_d)))))))))))).
