%   ORIGINAL: h4/finite__map/f__o__FUPDATE
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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/COND__ID: !t b. h4/bool/COND b t t = t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% 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/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/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y 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/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pred__set/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/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/pred__set/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/GSPEC__F: h4/pred__set/GSPEC (\x. h4/pair/_2C x F) = h4/pred__set/EMPTY
% 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/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/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/finite__map/FMERGE__DEF_c1: !x m g f. h4/finite__map/FAPPLY (h4/finite__map/FMERGE m f g) x = h4/bool/COND (~h4/bool/IN x (h4/finite__map/FDOM f)) (h4/finite__map/FAPPLY g x) (h4/bool/COND (~h4/bool/IN x (h4/finite__map/FDOM g)) (h4/finite__map/FAPPLY f x) (m (h4/finite__map/FAPPLY f x) (h4/finite__map/FAPPLY g x)))
% Assm: h4/finite__map/FDOM__FMERGE: !m g f. h4/finite__map/FDOM (h4/finite__map/FMERGE m f g) = h4/pred__set/UNION (h4/finite__map/FDOM f) (h4/finite__map/FDOM g)
% Assm: h4/finite__map/f__o__f__DEF_c0: !g f. h4/finite__map/FDOM (h4/finite__map/f__o__f f g) = h4/pred__set/INTER (h4/finite__map/FDOM g) (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (h4/finite__map/FAPPLY g x) (h4/finite__map/FDOM f))))
% Assm: h4/finite__map/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/finite__map/f__o__f__FEMPTY__2: !f. h4/finite__map/f__o__f f h4/finite__map/FEMPTY = h4/finite__map/FEMPTY
% 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/FUN__FMAP__EMPTY: !f. h4/finite__map/FUN__FMAP f h4/pred__set/EMPTY = h4/finite__map/FEMPTY
% Assm: h4/finite__map/FDOM__FMAP: !s f. h4/pred__set/FINITE s ==> h4/finite__map/FDOM (h4/finite__map/FUN__FMAP f s) = s
% Assm: h4/finite__map/f__o__DEF: !g f. h4/finite__map/f__o f g = h4/finite__map/f__o__f f (h4/finite__map/FUN__FMAP g (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (g x) (h4/finite__map/FDOM f)))))
% Assm: h4/finite__map/FDOM__f__o: !g f. h4/pred__set/FINITE (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (g x) (h4/finite__map/FDOM f)))) ==> h4/finite__map/FDOM (h4/finite__map/f__o f g) = h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (g x) (h4/finite__map/FDOM f)))
% Goal: !v k g fm. h4/pred__set/FINITE (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN (g x) (h4/finite__map/FDOM fm)))) /\ h4/pred__set/FINITE (h4/pred__set/GSPEC (\x. h4/pair/_2C x (g x = k))) ==> h4/finite__map/f__o (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) g = h4/finite__map/FMERGE (h4/combin/C h4/combin/K) (h4/finite__map/f__o fm g) (h4/finite__map/FUN__FMAP (h4/combin/K v) (h4/pred__set/GSPEC (\x. h4/pair/_2C x (g x = k))))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% 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_CONDu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% 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_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_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_combins_Ku_u_THM]: !y x. happ (happ h4/combin/K x) y = x
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (happ (h4/combin/C f) x) y = happ (happ f y) 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_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_predu_u_sets_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_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_predu_u_sets_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_GSPECu_u_F]: !_0. (!x. happ _0 x = h4/pair/_2C x F) ==> h4/pred__set/GSPEC _0 = h4/pred__set/EMPTY
% 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_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_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_finiteu_u_maps_FMERGEu_u_DEFu_c1]: !x m g f. ?v. (v <=> ~h4/bool/IN x (h4/finite__map/FDOM g)) /\ (?v'. (v' <=> ~h4/bool/IN x (h4/finite__map/FDOM f)) /\ h4/finite__map/FAPPLY (h4/finite__map/FMERGE m f g) x = h4/bool/COND v' (h4/finite__map/FAPPLY g x) (h4/bool/COND v (h4/finite__map/FAPPLY f x) (happ (happ m (h4/finite__map/FAPPLY f x)) (h4/finite__map/FAPPLY g x))))
% Assm [h4s_finiteu_u_maps_FDOMu_u_FMERGE]: !m g f. h4/finite__map/FDOM (h4/finite__map/FMERGE m f g) = h4/pred__set/UNION (h4/finite__map/FDOM f) (h4/finite__map/FDOM g)
% Assm [h4s_finiteu_u_maps_fu_u_ou_u_fu_u_DEFu_c0]: !_0. (!g f x. happ (happ (happ _0 g) f) x = h4/pair/_2C x (h4/bool/IN (h4/finite__map/FAPPLY g x) (h4/finite__map/FDOM f))) ==> (!g f. h4/finite__map/FDOM (h4/finite__map/f__o__f f g) = h4/pred__set/INTER (h4/finite__map/FDOM g) (h4/pred__set/GSPEC (happ (happ _0 g) f)))
% Assm [h4s_finiteu_u_maps_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_finiteu_u_maps_fu_u_ou_u_fu_u_FEMPTYu_u_2]: !f. h4/finite__map/f__o__f f h4/finite__map/FEMPTY = h4/finite__map/FEMPTY
% 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_FUNu_u_FMAPu_u_EMPTY]: !f. h4/finite__map/FUN__FMAP f h4/pred__set/EMPTY = h4/finite__map/FEMPTY
% Assm [h4s_finiteu_u_maps_FDOMu_u_FMAP]: !s f. h4/pred__set/FINITE s ==> h4/finite__map/FDOM (h4/finite__map/FUN__FMAP f s) = s
% Assm [h4s_finiteu_u_maps_fu_u_ou_u_DEF]: !_0. (!g f x. happ (happ (happ _0 g) f) x = h4/pair/_2C x (h4/bool/IN (happ g x) (h4/finite__map/FDOM f))) ==> (!g f. h4/finite__map/f__o f g = h4/finite__map/f__o__f f (h4/finite__map/FUN__FMAP g (h4/pred__set/GSPEC (happ (happ _0 g) f))))
% Assm [h4s_finiteu_u_maps_FDOMu_u_fu_u_o]: !_0. (!g f x. happ (happ (happ _0 g) f) x = h4/pair/_2C x (h4/bool/IN (happ g x) (h4/finite__map/FDOM f))) ==> (!g f. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 g) f)) ==> h4/finite__map/FDOM (h4/finite__map/f__o f g) = h4/pred__set/GSPEC (happ (happ _0 g) f))
% Goal: !_1. (!g k x. ?v. (v <=> happ g x = k) /\ happ (happ (happ _1 g) k) x = h4/pair/_2C x v) ==> (!_0. (!g fm x. happ (happ (happ _0 g) fm) x = h4/pair/_2C x (h4/bool/IN (happ g x) (h4/finite__map/FDOM fm))) ==> (!v k g fm. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 g) fm)) /\ h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _1 g) k)) ==> h4/finite__map/f__o (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) g = h4/finite__map/FMERGE (h4/combin/C h4/combin/K) (h4/finite__map/f__o fm g) (h4/finite__map/FUN__FMAP (happ h4/combin/K v) (h4/pred__set/GSPEC (happ (happ _1 g) k)))))
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_Q74936,TV_Q74932]: ![V_f, V_g]: (![V_x]: s(TV_Q74932,happ(s(t_fun(TV_Q74936,TV_Q74932),V_f),s(TV_Q74936,V_x))) = s(TV_Q74932,happ(s(t_fun(TV_Q74936,TV_Q74932),V_g),s(TV_Q74936,V_x))) => s(t_fun(TV_Q74936,TV_Q74932),V_f) = s(t_fun(TV_Q74936,TV_Q74932),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_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_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_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_CONDu_u_ID, axiom, ![TV_u_27a]: ![V_t, V_b]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_t),s(TV_u_27a,V_t))) = s(TV_u_27a,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_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_IMPu_u_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_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_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
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_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),h4s_combins_k),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(TV_u_27b,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_y))),s(TV_u_27b,V_x)))).
fof(ah4s_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_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_predu_u_sets_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_predu_u_sets_INu_u_INTER, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_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_GSPECu_u_F, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,f))) => s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
fof(ah4s_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_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_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_finiteu_u_maps_FMERGEu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_m, V_g, V_f]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_g)))))))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f)))))))) & s(TV_u_27a,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fmerge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_m),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_g))),s(TV_u_27b,V_x))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_vi_),s(TV_u_27a,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f),s(TV_u_27b,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_m),s(TV_u_27a,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f),s(TV_u_27b,V_x))))),s(TV_u_27a,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x)))))))))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_FMERGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_m, V_g, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_fmerge(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_m),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_g))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_union(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_27a),V_f))),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_g)))))).
fof(ah4s_finiteu_u_maps_fu_u_ou_u_fu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_g, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_bools_in(s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))) => ![V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))))).
fof(ah4s_finiteu_u_maps_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_finiteu_u_maps_fu_u_ou_u_fu_u_FEMPTYu_u_2, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fempty)).
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_FUNu_u_FMAPu_u_EMPTY, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty)).
fof(ah4s_finiteu_u_maps_FDOMu_u_FMAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_fun(TV_u_27a,t_bool),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_s))))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_finiteu_u_maps_fu_u_ou_u_DEF, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_g, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))) => ![V_g, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fu_u_o(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fu_u_ou_u_f(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_fu_u_o, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_g, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f))))))) => ![V_g, V_f]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f)))))))) => s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27c),h4s_finiteu_u_maps_fu_u_o(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27c),V_f)))))))).
fof(ch4s_finiteu_u_maps_fu_u_ou_u_FUPDATE, conjecture, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_g, V_k, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x))) = s(TV_u_27a,V_k)) & s(t_h4s_pairs_prod(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)))),V_uu_1),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27a,V_k))),s(TV_u_27c,V_x))) = s(t_h4s_pairs_prod(TV_u_27c,t_bool),h4s_pairs_u_2c(s(TV_u_27c,V_x),s(t_bool,V_v)))) => ![V_uu_0]: (![V_g, V_fm, V_x]: s(t_h4s_pairs_prod(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm))),s(TV_u_27c,V_x))) = s(t_h4s_pairs_prod(TV_u_27c,t_bool),h4s_pairs_u_2c(s(TV_u_27c,V_x),s(t_bool,h4s_bools_in(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,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_fm))))))) => ![V_v, V_k, V_g, V_fm]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm)))))))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)))),V_uu_1),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27a,V_k))))))))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_fu_u_o(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_fm),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27b,V_v))))),s(t_fun(TV_u_27c,TV_u_27a),V_g))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_fmerge(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b)),h4s_combins_c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b)),h4s_combins_k))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_fu_u_o(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_funu_u_fmap(s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27b)),h4s_combins_k),s(TV_u_27b,V_v))),s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27c,t_bool)))),V_uu_1),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27a,V_k))))))))))))).
