%   ORIGINAL: h4/fmaptree/apply__path__SNOC
% 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/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/rich__list/LIST__REL__APPEND__SING: !x2 x1 l2 l1 R. h4/list/LIST__REL R (h4/list/APPEND l1 (h4/list/CONS x1 h4/list/NIL)) (h4/list/APPEND l2 (h4/list/CONS x2 h4/list/NIL)) <=> h4/list/LIST__REL R l1 l2 /\ R x1 x2
% Assm: h4/fmaptree/apply__path__def_c1: !t h ft. h4/fmaptree/apply__path (h4/list/CONS h t) ft = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM (h4/fmaptree/map ft))) (h4/fmaptree/apply__path t (h4/finite__map/FAPPLY (h4/fmaptree/map ft) h)) h4/option/NONE
% Assm: h4/fmaptree/item__map__def: !ft. ft = h4/fmaptree/FTNode (h4/fmaptree/item ft) (h4/fmaptree/map ft)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/fmaptree/apply__path__def_c0: !ft. h4/fmaptree/apply__path h4/list/NIL ft = h4/option/SOME ft
% Assm: h4/fmaptree/applicable__paths__FINITE: !ft. h4/pred__set/FINITE (h4/pred__set/GSPEC (\p. h4/pair/_2C p (?ft_27. h4/fmaptree/apply__path p ft = h4/option/SOME ft_27)))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/fmaptree/fmaptree__nchotomy: !ft. ?i fm. ft = h4/fmaptree/FTNode i fm
% 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/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/map__thm: !i fm. h4/fmaptree/map (h4/fmaptree/FTNode i fm) = fm
% Assm: h4/option/option__case__ID: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/fmaptree/update__at__path__def_c0: !ft a. h4/fmaptree/update__at__path h4/list/NIL a ft = h4/option/SOME (h4/fmaptree/FTNode a (h4/fmaptree/map ft))
% Assm: h4/fmaptree/FTNode__11: !i2 i1 f2 f1. h4/fmaptree/FTNode i1 f1 = h4/fmaptree/FTNode i2 f2 <=> i1 = i2 /\ f1 = f2
% Assm: h4/rich__list/FOLDL__SINGLE: !x f e. h4/list/FOLDL f e (h4/list/CONS x h4/list/NIL) = f e x
% Assm: h4/rich__list/SCANR0_c0: !f e. h4/rich__list/SCANR f e h4/list/NIL = h4/list/CONS e h4/list/NIL
% Assm: h4/list/SINGL__LIST__APPLY__L: !x f. h4/list/LIST__BIND (h4/list/CONS x h4/list/NIL) f = f x
% Assm: h4/rich__list/SCANL0_c0: !f e. h4/rich__list/SCANL f e h4/list/NIL = h4/list/CONS e h4/list/NIL
% Assm: h4/option/option__Axiom: !f e. ?fn. fn h4/option/NONE = e /\ (!x. fn (h4/option/SOME x) = f x)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/rich__list/EVERY2__APPEND__suff: !l4 l3 l2 l1 R. h4/list/LIST__REL R l1 l2 /\ h4/list/LIST__REL R l3 l4 ==> h4/list/LIST__REL R (h4/list/APPEND l1 l3) (h4/list/APPEND l2 l4)
% Assm: h4/rich__list/EVERY2__APPEND: !l4 l3 l2 l1 R. h4/list/LIST__REL R l1 l2 /\ h4/list/LIST__REL R l3 l4 <=> h4/list/LIST__REL R (h4/list/APPEND l1 l3) (h4/list/APPEND l2 l4) /\ h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ h4/list/LENGTH l3 = h4/list/LENGTH l4
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/list/LENGTH__APPEND: !l2 l1. h4/list/LENGTH (h4/list/APPEND l1 l2) = h4/arithmetic/_2B (h4/list/LENGTH l1) (h4/list/LENGTH l2)
% Assm: h4/list/LIST__REL__def_c0: !R. h4/list/LIST__REL R h4/list/NIL h4/list/NIL <=> T
% Assm: h4/list/LIST__REL__def_c3: !bs b as a R. h4/list/LIST__REL R (h4/list/CONS a as) (h4/list/CONS b bs) <=> R a b /\ h4/list/LIST__REL R as bs
% Assm: h4/list/APPEND__ASSOC: !l3 l2 l1. h4/list/APPEND l1 (h4/list/APPEND l2 l3) = h4/list/APPEND (h4/list/APPEND l1 l2) l3
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/numeral/numeral__distrib_c13: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm: h4/list/FOLDL0_c1: !x l f e. h4/list/FOLDL f e (h4/list/CONS x l) = h4/list/FOLDL f (f e x) l
% Assm: h4/list/LIST__BIND__THM_c1: !t h f. h4/list/LIST__BIND (h4/list/CONS h t) f = h4/list/APPEND (f h) (h4/list/LIST__BIND t f)
% Assm: h4/list/LIST__BIND__THM_c0: !f. h4/list/LIST__BIND h4/list/NIL f = h4/list/NIL
% Assm: h4/list/APPEND__NIL: !l. h4/list/APPEND l h4/list/NIL = l
% Assm: h4/finite__map/FLOOKUP__o__f: !k fm f. h4/finite__map/FLOOKUP (h4/finite__map/o__f f fm) k = h4/option/option__CASE (h4/finite__map/FLOOKUP fm k) h4/option/NONE (\v. h4/option/SOME (f v))
% Assm: h4/finite__map/FLOOKUP__FUNION: !k f2 f1. h4/finite__map/FLOOKUP (h4/finite__map/FUNION f1 f2) k = h4/option/option__CASE (h4/finite__map/FLOOKUP f1 k) (h4/finite__map/FLOOKUP f2 k) (\v. h4/option/SOME v)
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/option__case__cong: !v_27 v f_27 f M_27 M. M = M_27 /\ (M_27 = h4/option/NONE ==> v = v_27) /\ (!x. M_27 = h4/option/SOME x ==> f x = f_27 x) ==> h4/option/option__CASE M v f = h4/option/option__CASE M_27 v_27 f_27
% Assm: h4/option/option__CLAUSES_c14: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__CLAUSES_c9: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% 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/fupd__at__path__def_c0: !ft f. h4/fmaptree/fupd__at__path h4/list/NIL f ft = f ft
% 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/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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% 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/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% 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/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% 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/FDOM__FINITE: !fm. h4/pred__set/FINITE (h4/finite__map/FDOM fm)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/list/NOT__NIL__CONS: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/pred__set/FINITE__BIGUNION__EQ: !P. h4/pred__set/FINITE (h4/pred__set/BIGUNION P) <=> h4/pred__set/FINITE P /\ (!s. h4/bool/IN s P ==> h4/pred__set/FINITE s)
% Assm: h4/pred__set/IN__BIGUNION: !x sos. h4/bool/IN x (h4/pred__set/BIGUNION sos) <=> (?s. h4/bool/IN x s /\ h4/bool/IN s sos)
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/pred__set/IMAGE__FINITE: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm: h4/pred__set/SING0: !x. h4/pred__set/SING (h4/pred__set/INSERT 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/pred__set/FINITE__UNION: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm: h4/pred__set/SING__FINITE: !s. h4/pred__set/SING s ==> h4/pred__set/FINITE s
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c2: !x X P. h4/bool/COND P X h4/option/NONE = h4/option/SOME x <=> P /\ X = h4/option/SOME x
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/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/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ 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/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ 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/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/finite__map/FDOM__o__f: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm: h4/finite__map/o__f__FAPPLY: !x g f. h4/bool/IN x (h4/finite__map/FDOM g) ==> h4/finite__map/FAPPLY (h4/finite__map/o__f f g) x = f (h4/finite__map/FAPPLY g x)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/finite__map/FUNION__DEF_c1: !x g f. h4/finite__map/FAPPLY (h4/finite__map/FUNION f g) x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/finite__map/FAPPLY f x) (h4/finite__map/FAPPLY g x)
% Assm: h4/finite__map/FUNION__DEF_c0: !g f. h4/finite__map/FDOM (h4/finite__map/FUNION f g) = h4/pred__set/UNION (h4/finite__map/FDOM f) (h4/finite__map/FDOM g)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Goal: !x p ft. h4/fmaptree/apply__path (h4/list/APPEND p (h4/list/CONS x h4/list/NIL)) ft = h4/option/option__CASE (h4/fmaptree/apply__path p ft) h4/option/NONE (\ft_27. h4/finite__map/FLOOKUP (h4/fmaptree/map ft_27) x)
%   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_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_richu_u_lists_LISTu_u_RELu_u_APPENDu_u_SING]: !x2 x1 l2 l1 R. h4/list/LIST__REL R (h4/list/APPEND l1 (h4/list/CONS x1 h4/list/NIL)) (h4/list/APPEND l2 (h4/list/CONS x2 h4/list/NIL)) <=> h4/list/LIST__REL R l1 l2 /\ happ (happ R x1) x2
% Assm [h4s_fmaptrees_applyu_u_pathu_u_defu_c1]: !t h ft. h4/fmaptree/apply__path (h4/list/CONS h t) ft = h4/bool/COND (h4/bool/IN h (h4/finite__map/FDOM (h4/fmaptree/map ft))) (h4/fmaptree/apply__path t (h4/finite__map/FAPPLY (h4/fmaptree/map ft) h)) h4/option/NONE
% Assm [h4s_fmaptrees_itemu_u_mapu_u_def]: !ft. ft = h4/fmaptree/FTNode (h4/fmaptree/item ft) (h4/fmaptree/map ft)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_fmaptrees_applyu_u_pathu_u_defu_c0]: !ft. h4/fmaptree/apply__path h4/list/NIL ft = happ h4/option/SOME ft
% Assm [h4s_fmaptrees_applicableu_u_pathsu_u_FINITE]: !_0. (!ft p. ?v. (v <=> (?ft_27. h4/fmaptree/apply__path p ft = happ h4/option/SOME ft_27)) /\ happ (happ _0 ft) p = h4/pair/_2C p v) ==> (!ft. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ _0 ft)))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_fmaptrees_fmaptreeu_u_nchotomy]: !ft. ?i fm. ft = h4/fmaptree/FTNode i fm
% Assm [h4s_fmaptrees_updateu_u_atu_u_pathu_u_defu_c1]: !_0. (!ft h ft_27. happ (happ (happ _0 ft) h) ft_27 = happ 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_fupdu_u_atu_u_pathu_u_defu_c1]: !_0. (!ft h ft_27. happ (happ (happ _0 ft) h) ft_27 = happ 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_mapu_u_thm]: !i fm. h4/fmaptree/map (h4/fmaptree/FTNode i fm) = fm
% Assm [h4s_options_optionu_u_caseu_u_ID]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_fmaptrees_updateu_u_atu_u_pathu_u_defu_c0]: !ft a. h4/fmaptree/update__at__path h4/list/NIL a ft = happ h4/option/SOME (h4/fmaptree/FTNode a (h4/fmaptree/map ft))
% Assm [h4s_fmaptrees_FTNodeu_u_11]: !i2 i1 f2 f1. h4/fmaptree/FTNode i1 f1 = h4/fmaptree/FTNode i2 f2 <=> i1 = i2 /\ f1 = f2
% Assm [h4s_richu_u_lists_FOLDLu_u_SINGLE]: !x f e. h4/list/FOLDL f e (h4/list/CONS x h4/list/NIL) = happ (happ f e) x
% Assm [h4s_richu_u_lists_SCANR0u_c0]: !f e. h4/rich__list/SCANR f e h4/list/NIL = h4/list/CONS e h4/list/NIL
% Assm [h4s_lists_SINGLu_u_LISTu_u_APPLYu_u_L]: !x f. h4/list/LIST__BIND (h4/list/CONS x h4/list/NIL) f = happ f x
% Assm [h4s_richu_u_lists_SCANL0u_c0]: !f e. h4/rich__list/SCANL f e h4/list/NIL = h4/list/CONS e h4/list/NIL
% Assm [h4s_options_optionu_u_Axiom]: !f e. ?fn. happ fn h4/option/NONE = e /\ (!x. happ fn (happ h4/option/SOME x) = happ f x)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_richu_u_lists_EVERY2u_u_APPENDu_u_suff]: !l4 l3 l2 l1 R. h4/list/LIST__REL R l1 l2 /\ h4/list/LIST__REL R l3 l4 ==> h4/list/LIST__REL R (h4/list/APPEND l1 l3) (h4/list/APPEND l2 l4)
% Assm [h4s_richu_u_lists_EVERY2u_u_APPEND]: !l4 l3 l2 l1 R. h4/list/LIST__REL R l1 l2 /\ h4/list/LIST__REL R l3 l4 <=> h4/list/LIST__REL R (h4/list/APPEND l1 l3) (h4/list/APPEND l2 l4) /\ h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ h4/list/LENGTH l3 = h4/list/LENGTH l4
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_lists_LENGTHu_u_APPEND]: !l2 l1. h4/list/LENGTH (h4/list/APPEND l1 l2) = h4/arithmetic/_2B (h4/list/LENGTH l1) (h4/list/LENGTH l2)
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c0]: !R. h4/list/LIST__REL R h4/list/NIL h4/list/NIL <=> T
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c3]: !bs b as a R. h4/list/LIST__REL R (h4/list/CONS a as) (h4/list/CONS b bs) <=> happ (happ R a) b /\ h4/list/LIST__REL R as bs
% Assm [h4s_lists_APPENDu_u_ASSOC]: !l3 l2 l1. h4/list/APPEND l1 (h4/list/APPEND l2 l3) = h4/list/APPEND (h4/list/APPEND l1 l2) l3
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_numerals_numeralu_u_distribu_c13]: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (happ h4/option/SOME x) v f = happ f x
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_lists_FOLDL0u_c0]: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm [h4s_lists_FOLDL0u_c1]: !x l f e. h4/list/FOLDL f e (h4/list/CONS x l) = h4/list/FOLDL f (happ (happ f e) x) l
% Assm [h4s_lists_LISTu_u_BINDu_u_THMu_c1]: !t h f. h4/list/LIST__BIND (h4/list/CONS h t) f = h4/list/APPEND (happ f h) (h4/list/LIST__BIND t f)
% Assm [h4s_lists_LISTu_u_BINDu_u_THMu_c0]: !f. h4/list/LIST__BIND h4/list/NIL f = h4/list/NIL
% Assm [h4s_lists_APPENDu_u_NIL]: !l. h4/list/APPEND l h4/list/NIL = l
% Assm [h4s_finiteu_u_maps_FLOOKUPu_u_ou_u_f]: !_0. (!f v. happ (happ _0 f) v = happ h4/option/SOME (happ f v)) ==> (!k fm f. h4/finite__map/FLOOKUP (h4/finite__map/o__f f fm) k = h4/option/option__CASE (h4/finite__map/FLOOKUP fm k) h4/option/NONE (happ _0 f))
% Assm [h4s_finiteu_u_maps_FLOOKUPu_u_FUNION]: !_0. (!v. happ _0 v = happ h4/option/SOME v) ==> (!k f2 f1. h4/finite__map/FLOOKUP (h4/finite__map/FUNION f1 f2) k = h4/option/option__CASE (h4/finite__map/FLOOKUP f1 k) (h4/finite__map/FLOOKUP f2 k) _0)
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = happ h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. happ h4/option/SOME x = happ h4/option/SOME y <=> x = y
% Assm [h4s_options_optionu_u_caseu_u_cong]: !v_27 v f_27 f M_27 M. M = M_27 /\ (M_27 = h4/option/NONE ==> v = v_27) /\ (!x. M_27 = happ h4/option/SOME x ==> happ f x = happ f_27 x) ==> h4/option/option__CASE M v f = h4/option/option__CASE M_27 v_27 f_27
% Assm [h4s_options_optionu_u_CLAUSESu_c14]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_CLAUSESu_c9]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% 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_fupdu_u_atu_u_pathu_u_defu_c0]: !ft f. h4/fmaptree/fupd__at__path h4/list/NIL f ft = happ f ft
% 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_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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% 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_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% 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)) (happ h4/option/SOME (h4/finite__map/FAPPLY f x)) h4/option/NONE
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% 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_FDOMu_u_FINITE]: !fm. h4/pred__set/FINITE (h4/finite__map/FDOM fm)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_lists_NOTu_u_NILu_u_CONS]: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_predu_u_sets_FINITEu_u_BIGUNIONu_u_EQ]: !P. h4/pred__set/FINITE (h4/pred__set/BIGUNION P) <=> h4/pred__set/FINITE P /\ (!s. h4/bool/IN s P ==> h4/pred__set/FINITE s)
% Assm [h4s_predu_u_sets_INu_u_BIGUNION]: !x sos. h4/bool/IN x (h4/pred__set/BIGUNION sos) <=> (?s. h4/bool/IN x s /\ h4/bool/IN s sos)
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_predu_u_sets_IMAGEu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm [h4s_predu_u_sets_SING0]: !x. h4/pred__set/SING (h4/pred__set/INSERT 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_predu_u_sets_FINITEu_u_UNION]: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm [h4s_predu_u_sets_SINGu_u_FINITE]: !s. h4/pred__set/SING s ==> h4/pred__set/FINITE s
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c2]: !x X P. h4/bool/COND P X h4/option/NONE = happ h4/option/SOME x <=> P /\ X = happ h4/option/SOME x
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_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_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_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ 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_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ 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_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_finiteu_u_maps_FDOMu_u_ou_u_f]: !g f. h4/finite__map/FDOM (h4/finite__map/o__f f g) = h4/finite__map/FDOM g
% Assm [h4s_finiteu_u_maps_ou_u_fu_u_FAPPLY]: !x g f. h4/bool/IN x (h4/finite__map/FDOM g) ==> h4/finite__map/FAPPLY (h4/finite__map/o__f f g) x = happ f (h4/finite__map/FAPPLY g x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_finiteu_u_maps_FUNIONu_u_DEFu_c1]: !x g f. h4/finite__map/FAPPLY (h4/finite__map/FUNION f g) x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/finite__map/FAPPLY f x) (h4/finite__map/FAPPLY g x)
% Assm [h4s_finiteu_u_maps_FUNIONu_u_DEFu_c0]: !g f. h4/finite__map/FDOM (h4/finite__map/FUNION f g) = h4/pred__set/UNION (h4/finite__map/FDOM f) (h4/finite__map/FDOM g)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = happ h4/option/SOME x)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Goal: !_0. (!x ft_27. happ (happ _0 x) ft_27 = h4/finite__map/FLOOKUP (h4/fmaptree/map ft_27) x) ==> (!x p ft. h4/fmaptree/apply__path (h4/list/APPEND p (h4/list/CONS x h4/list/NIL)) ft = h4/option/option__CASE (h4/fmaptree/apply__path p ft) h4/option/NONE (happ _0 x))
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_Q1348494,TV_Q1348490]: ![V_f, V_g]: (![V_x]: s(TV_Q1348490,happ(s(t_fun(TV_Q1348494,TV_Q1348490),V_f),s(TV_Q1348494,V_x))) = s(TV_Q1348490,happ(s(t_fun(TV_Q1348494,TV_Q1348490),V_g),s(TV_Q1348494,V_x))) => s(t_fun(TV_Q1348494,TV_Q1348490),V_f) = s(t_fun(TV_Q1348494,TV_Q1348490),V_g))).
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_richu_u_lists_LISTu_u_RELu_u_APPENDu_u_SING, axiom, ![TV_u_27a,TV_u_27b]: ![V_x2, V_x1, V_l2, V_l1, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),V_l2),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_x2),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))))))) <=> (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2)))) & 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_x1))),s(TV_u_27b,V_x2))))))).
fof(ah4s_fmaptrees_applyu_u_pathu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_h, V_ft]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_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_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_fmaptrees_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),V_t),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)))).
fof(ah4s_fmaptrees_itemu_u_mapu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_ft]: 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,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_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_fmaptrees_applyu_u_pathu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_ft]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),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)),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))),h4s_options_some),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft)))).
fof(ah4s_fmaptrees_applicableu_u_pathsu_u_FINITE, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_ft, V_p]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_ftu_27]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),V_p),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)),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))),h4s_options_some),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ftu_27)))) & s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_bool))),V_uu_0),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft))),s(t_h4s_lists_list(TV_u_27a),V_p))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_bool),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_p),s(t_bool,V_v)))) => ![V_ft]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_bool))),V_uu_0),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft)))))))))).
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_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_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_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)),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))),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_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)),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))),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_mapu_u_thm, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_fm]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),h4s_fmaptrees_map(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(t_h4s_finiteu_u_maps_fmap(TV_u_27b,t_h4s_fmaptrees_fmaptree(TV_u_27b,TV_u_27a)),V_fm)).
fof(ah4s_options_optionu_u_caseu_u_ID, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_fmaptrees_updateu_u_atu_u_pathu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![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_nil),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)),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))),h4s_options_some),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)),h4s_fmaptrees_map(s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft)))))))).
fof(ah4s_fmaptrees_FTNodeu_u_11, axiom, ![TV_u_27a,TV_u_27b]: ![V_i2, V_i1, V_f2, V_f1]: (s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,V_i1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_f1))) = s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),h4s_fmaptrees_ftnode(s(TV_u_27b,V_i2),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_f2))) <=> (s(TV_u_27b,V_i1) = s(TV_u_27b,V_i2) & s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_f1) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),V_f2)))).
fof(ah4s_richu_u_lists_FOLDLu_u_SINGLE, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_e]: s(TV_u_27a,h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_e),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_x),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(TV_u_27b,V_x)))).
fof(ah4s_richu_u_lists_SCANR0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(t_h4s_lists_list(TV_u_27b),h4s_richu_u_lists_scanr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))).
fof(ah4s_lists_SINGLu_u_LISTu_u_APPLYu_u_L, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_x),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f),s(TV_u_27b,V_x)))).
fof(ah4s_richu_u_lists_SCANL0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(t_h4s_lists_list(TV_u_27b),h4s_richu_u_lists_scanl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))).
fof(ah4s_options_optionu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(TV_u_27b,V_e) & ![V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_EQu_u_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_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_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_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_richu_u_lists_EVERY2u_u_APPENDu_u_suff, axiom, ![TV_u_27a,TV_u_27b]: ![V_l4, V_l3, V_l2, V_l1, V_R]: ((p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_l3),s(t_h4s_lists_list(TV_u_27b),V_l4))))) => p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l3))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),V_l2),s(t_h4s_lists_list(TV_u_27b),V_l4)))))))).
fof(ah4s_richu_u_lists_EVERY2u_u_APPEND, axiom, ![TV_u_27a,TV_u_27b]: ![V_l4, V_l3, V_l2, V_l1, V_R]: ((p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_l3),s(t_h4s_lists_list(TV_u_27b),V_l4))))) <=> (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l3))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),V_l2),s(t_h4s_lists_list(TV_u_27b),V_l4)))))) & (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l2))) & s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l3))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l4))))))).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_lists_LENGTHu_u_APPEND, axiom, ![TV_u_27a]: ![V_l2, V_l1]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R]: s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_bs, V_b, V_as, V_a, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a),s(t_h4s_lists_list(TV_u_27a),V_as))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_b),s(t_h4s_lists_list(TV_u_27b),V_bs)))))) <=> (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_a))),s(TV_u_27b,V_b)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_as),s(t_h4s_lists_list(TV_u_27b),V_bs))))))).
fof(ah4s_lists_APPENDu_u_ASSOC, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l2),s(t_h4s_lists_list(TV_u_27a),V_l3))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))),s(t_h4s_lists_list(TV_u_27a),V_l3)))).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_numerals_numeralu_u_distribu_c13, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_FOLDL0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_lists_FOLDL0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e))),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_lists_LISTu_u_BINDu_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_h),s(t_h4s_lists_list(TV_u_27b),V_t))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),V_t),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f)))))).
fof(ah4s_lists_LISTu_u_BINDu_u_THMu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_APPENDu_u_NIL, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_finiteu_u_maps_FLOOKUPu_u_ou_u_f, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_0]: (![V_f, V_v]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27c,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_h4s_options_option(TV_u_27a))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_f))),s(TV_u_27c,V_v))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_f),s(TV_u_27c,V_v))))) => ![V_k, V_fm, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27c,TV_u_27a),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_fm))),s(TV_u_27b,V_k))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27c),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_fm),s(TV_u_27b,V_k))),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27c,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_h4s_options_option(TV_u_27a))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_f))))))).
fof(ah4s_finiteu_u_maps_FLOOKUPu_u_FUNION, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_v]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),V_uu_0),s(TV_u_27a,V_v))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_v))) => ![V_k, V_f2, V_f1]: s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f2))),s(TV_u_27b,V_k))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f1),s(TV_u_27b,V_k))),s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f2),s(TV_u_27b,V_k))),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),V_uu_0))))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,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),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_optionu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b]: ![V_vu_27, V_v, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_options_option(TV_u_27a),V_M) = s(t_h4s_options_option(TV_u_27a),V_Mu_27) & ((s(t_h4s_options_option(TV_u_27a),V_Mu_27) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) => s(TV_u_27b,V_v) = s(TV_u_27b,V_vu_27)) & ![V_x]: (s(t_h4s_options_option(TV_u_27a),V_Mu_27) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),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))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_fu_27),s(TV_u_27a,V_x)))))) => s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_M),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_Mu_27),s(TV_u_27b,V_vu_27),s(t_fun(TV_u_27a,TV_u_27b),V_fu_27))))).
fof(ah4s_options_optionu_u_CLAUSESu_c14, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_CLAUSESu_c9, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
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_fupdu_u_atu_u_pathu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![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_nil),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)),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))),V_f),s(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b),V_ft)))).
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_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_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_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_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),happ(s(t_fun(TV_u_27b,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_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_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_FDOMu_u_FINITE, axiom, ![TV_u_27a,TV_u_27b]: ![V_fm]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm))))))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_NOTu_u_NILu_u_CONS, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0u_27),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_predu_u_sets_FINITEu_u_BIGUNIONu_u_EQ, axiom, ![TV_u_27a]: ![V_P]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))) <=> (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_INu_u_BIGUNION, axiom, ![TV_u_27a]: ![V_x, V_sos]: (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_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sos)))))) <=> ?[V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sos))))))).
fof(ah4s_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_lists_listu_u_nchotomy, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) | ?[V_h, V_t]: s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))).
fof(ah4s_predu_u_sets_IMAGEu_u_FINITE, axiom, ![TV_u_27b,TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_SING0, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_predu_u_sets_sing(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))))).
fof(ah4s_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_predu_u_sets_FINITEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_SINGu_u_FINITE, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_sing(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_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_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_x, 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),V_X),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) <=> (p(s(t_bool,V_P)) & s(t_h4s_options_option(TV_u_27a),V_X) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x)))))).
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_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_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_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_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_dcu_u_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_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_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_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_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_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_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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
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_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_bools_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = 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_finiteu_u_maps_FDOMu_u_ou_u_f, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))).
fof(ah4s_finiteu_u_maps_ou_u_fu_u_FAPPLY, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))))) => s(TV_u_27c,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_ou_u_f(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_finiteu_u_maps_FUNIONu_u_DEFu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_finiteu_u_maps_FUNIONu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))))).
fof(ah4s_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_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),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))).
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(ch4s_fmaptrees_applyu_u_pathu_u_SNOC, conjecture, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_x, 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)))),V_uu_0),s(TV_u_27a,V_x))),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_finiteu_u_maps_flookup(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_ftu_27))),s(TV_u_27a,V_x))) => ![V_x, V_p, V_ft]: s(t_h4s_options_option(t_h4s_fmaptrees_fmaptree(TV_u_27a,TV_u_27b)),h4s_fmaptrees_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_p),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))),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_applyu_u_path(s(t_h4s_lists_list(TV_u_27a),V_p),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_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)))),V_uu_0),s(TV_u_27a,V_x))))))).
