%   ORIGINAL: h4/pred__set/SURJ__COMPOSE
% 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/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/pred__set/INJ__DEF: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> f x = f y ==> x = y)
% Assm: h4/pred__set/INJ__COMPOSE: !u t s g f. h4/pred__set/INJ f s t /\ h4/pred__set/INJ g t u ==> h4/pred__set/INJ (h4/combin/o g f) s u
% Assm: h4/pred__set/SURJ__DEF: !t s f. h4/pred__set/SURJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x. h4/bool/IN x t ==> (?y. h4/bool/IN y s /\ f y = x))
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pred__set/IMAGE__COMPOSE: !s g f. h4/pred__set/IMAGE (h4/combin/o f g) s = h4/pred__set/IMAGE f (h4/pred__set/IMAGE g s)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/pred__set/SURJ__ID: !s. h4/pred__set/SURJ (\x. x) s s
% Assm: h4/sum/sum__axiom: !g f. h4/bool/_3F_21 (\h. h4/combin/o h h4/sum/INL = f /\ h4/combin/o h h4/sum/INR = g)
% 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/sum/sum__case__cong: !f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!x. M_27 = h4/sum/INL x ==> f x = f_27 x) /\ (!y. M_27 = h4/sum/INR y ==> f1 y = f1_27 y) ==> h4/sum/sum__CASE M f f1 = h4/sum/sum__CASE M_27 f_27 f1_27
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/sum/sum__CASES: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/option/OPTION__APPLY__o: !x g f. h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/SOME h4/combin/o) f) g) x = h4/option/OPTION__APPLY f (h4/option/OPTION__APPLY g x)
% Assm: h4/sum/INL__11: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/sum/INL__DEF: !e. h4/sum/INL e = h4/sum/ABS__sum (\b x y. x = e /\ b)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% 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/sum/INR__neq__INL: !v2 v1. ~(h4/sum/INR v2 = h4/sum/INL v1)
% Assm: h4/sum/sum__ISO__DEF_c0: !a. h4/sum/ABS__sum (h4/sum/REP__sum a) = a
% Assm: h4/sum/INR__DEF: !e. h4/sum/INR e = h4/sum/ABS__sum (\b x y. y = e /\ ~b)
% Assm: h4/sum/sum__ISO__DEF_c1: !r. h4/sum/IS__SUM__REP r <=> h4/sum/REP__sum (h4/sum/ABS__sum r) = r
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/sum/INR__11: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/bool/SELECT__REFL__2: !x. h4/min/_40 (\y. x = y) = x
% Assm: h4/sum/IS__SUM__REP0: !f. h4/sum/IS__SUM__REP f <=> (?v1 v2. f = (\b x y. x = v1 /\ b) \/ f = (\b x y. y = v2 /\ ~b))
% 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/pred__set/IMAGE__DEF: !s f. h4/pred__set/IMAGE f s = h4/pred__set/GSPEC (\x. h4/pair/_2C (f x) (h4/bool/IN x s))
% Assm: h4/combin/o__ABS__R: !g f. h4/combin/o f (\x. g x) = (\x. f (g x))
% Assm: h4/combin/GEN__literal__case__RAND: !v f P. P (h4/bool/literal__case f v) = h4/bool/literal__case (h4/combin/o P f) v
% Assm: h4/combin/GEN__LET__RAND: !v f P. P (h4/bool/LET f v) = h4/bool/LET (h4/combin/o P f) v
% Assm: h4/combin/o__ASSOC: !h g f. h4/combin/o f (h4/combin/o g h) = h4/combin/o (h4/combin/o f g) h
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/option/OPTION__MAP__COMPOSE: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% Assm: h4/combin/K__o__THM_c0: !v f. h4/combin/o (h4/combin/K v) f = h4/combin/K v
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% 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__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/pred__set/INJ__IFF: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> (f x = f y <=> x = y))
% Assm: h4/while/HOARE__SPEC__DEF: !Q P C. h4/while/HOARE__SPEC P C Q <=> (!s. P s ==> Q (C s))
% Assm: h4/pred__set/INJ__INSERT: !x t s f. h4/pred__set/INJ f (h4/pred__set/INSERT x s) t <=> h4/pred__set/INJ f s t /\ h4/bool/IN (f x) t /\ (!y. h4/bool/IN y s /\ f x = f y ==> x = y)
% Assm: h4/pred__set/INJ__DELETE: !t s f. h4/pred__set/INJ f s t ==> (!e. h4/bool/IN e s ==> h4/pred__set/INJ f (h4/pred__set/DELETE s e) (h4/pred__set/DELETE t (f e)))
% Assm: h4/pred__set/IMAGE__IN: !x s. h4/bool/IN x s ==> (!f. h4/bool/IN (f x) (h4/pred__set/IMAGE f s))
% Assm: h4/pred__set/IMAGE__INSERT: !x s f. h4/pred__set/IMAGE f (h4/pred__set/INSERT x s) = h4/pred__set/INSERT (f x) (h4/pred__set/IMAGE f s)
% Assm: h4/pred__set/IMAGE__11: !s2 s1 f. (!x y. f x = f y <=> x = y) ==> (h4/pred__set/IMAGE f s1 = h4/pred__set/IMAGE f s2 <=> s1 = s2)
% Assm: h4/pred__set/IMAGE__CONG: !s_27 s f_27 f. s = s_27 /\ (!x. h4/bool/IN x s_27 ==> f x = f_27 x) ==> h4/pred__set/IMAGE f s = h4/pred__set/IMAGE f_27 s_27
% Assm: h4/pred__set/IMAGE__DELETE: !x s f. ~h4/bool/IN x s ==> h4/pred__set/IMAGE f (h4/pred__set/DELETE s x) = h4/pred__set/IMAGE f s
% Assm: h4/bool/literal__case__THM: !x f. h4/bool/literal__case f x = f x
% Assm: h4/pred__set/IMAGE__INTER: !t s f. h4/pred__set/SUBSET (h4/pred__set/IMAGE f (h4/pred__set/INTER s t)) (h4/pred__set/INTER (h4/pred__set/IMAGE f s) (h4/pred__set/IMAGE f t))
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/relation/TFL__INDUCTIVE__INVARIANT__ON__WFREC: !x f R P M D. f = h4/relation/WFREC R M /\ h4/relation/WF R /\ h4/relation/INDUCTIVE__INVARIANT__ON R D P M /\ D x ==> P x (f x)
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/pred__set/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/option/OPTION__APPLY__def_c1: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm: h4/option/OPTION__APPLY__def_c0: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% 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/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/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/IN__DELETE: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/pred__set/DIFF__INSERT: !x t s. h4/pred__set/DIFF s (h4/pred__set/INSERT x t) = h4/pred__set/DIFF (h4/pred__set/DELETE s x) t
% Assm: h4/pred__set/DELETE__DEF: !x s. h4/pred__set/DELETE s x = h4/pred__set/DIFF s (h4/pred__set/INSERT x h4/pred__set/EMPTY)
% Assm: h4/pred__set/IN__DIFF: !x t s. h4/bool/IN x (h4/pred__set/DIFF s t) <=> h4/bool/IN x s /\ ~h4/bool/IN x t
% Assm: h4/pred__set/DIFF__EMPTY: !s. h4/pred__set/DIFF s h4/pred__set/EMPTY = s
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/relation/INDUCTIVE__INVARIANT__ON__WFREC: !x R P M D. h4/relation/WF R /\ h4/relation/INDUCTIVE__INVARIANT__ON R D P M /\ D x ==> P x (h4/relation/WFREC R M x)
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Goal: !u t s g f. h4/pred__set/SURJ f s t /\ h4/pred__set/SURJ g t u ==> h4/pred__set/SURJ (h4/combin/o g f) s u
%   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_combins_ou_u_THM]: !x g f. happ (happ (happ h4/combin/o f) g) x = happ f (happ g x)
% Assm [h4s_predu_u_sets_INJu_u_DEF]: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> happ f x = happ f y ==> x = y)
% Assm [h4s_predu_u_sets_INJu_u_COMPOSE]: !u t s g f. h4/pred__set/INJ f s t /\ h4/pred__set/INJ g t u ==> h4/pred__set/INJ (happ (happ h4/combin/o g) f) s u
% Assm [h4s_predu_u_sets_SURJu_u_DEF]: !t s f. h4/pred__set/SURJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x. h4/bool/IN x t ==> (?y. h4/bool/IN y s /\ happ f y = x))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_predu_u_sets_IMAGEu_u_COMPOSE]: !s g f. h4/pred__set/IMAGE (happ (happ h4/combin/o f) g) s = h4/pred__set/IMAGE f (h4/pred__set/IMAGE g s)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_predu_u_sets_SURJu_u_ID]: !_0. (!x. happ _0 x = x) ==> (!s. h4/pred__set/SURJ _0 s s)
% Assm [h4s_sums_sumu_u_axiom]: !_0. (!f g h. happ (happ (happ _0 f) g) h <=> happ (happ h4/combin/o h) h4/sum/INL = f /\ happ (happ h4/combin/o h) h4/sum/INR = g) ==> (!g f. h4/bool/_3F_21 (happ (happ _0 f) g))
% 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_sums_sumu_u_caseu_u_cong]: !f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!x. M_27 = happ h4/sum/INL x ==> happ f x = happ f_27 x) /\ (!y. M_27 = happ h4/sum/INR y ==> happ f1 y = happ f1_27 y) ==> h4/sum/sum__CASE M f f1 = h4/sum/sum__CASE M_27 f_27 f1_27
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (happ h4/sum/INR y) f f1 = happ f1 y
% Assm [h4s_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (happ h4/sum/INL x) f f1 = happ f x
% Assm [h4s_sums_sumu_u_CASES]: !ss. (?x. ss = happ h4/sum/INL x) \/ (?y. ss = happ h4/sum/INR y)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (happ (happ h4/combin/o f) g) x = happ f (happ g x)
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_options_OPTIONu_u_APPLYu_u_o]: !x g f. h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/SOME h4/combin/o) f) g) x = h4/option/OPTION__APPLY f (h4/option/OPTION__APPLY g x)
% Assm [h4s_sums_INLu_u_11]: !y x. happ h4/sum/INL x = happ h4/sum/INL y <=> x = y
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_sums_INLu_u_DEF]: !_2. (!x e b y. happ (happ (happ (happ _2 x) e) b) y <=> x = e /\ b) ==> (!_1. (!e b x. happ (happ (happ _1 e) b) x = happ (happ (happ _2 x) e) b) ==> (!_0. (!e b. happ (happ _0 e) b = happ (happ _1 e) b) ==> (!e. happ h4/sum/INL e = h4/sum/ABS__sum (happ _0 e))))
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% 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_sums_INRu_u_nequ_u_INL]: !v2 v1. ~(happ h4/sum/INR v2 = happ h4/sum/INL v1)
% Assm [h4s_sums_sumu_u_ISOu_u_DEFu_c0]: !a. h4/sum/ABS__sum (h4/sum/REP__sum a) = a
% Assm [h4s_sums_INRu_u_DEF]: !_2. (!e b y. happ (happ (happ _2 e) b) y <=> y = e /\ ~b) ==> (!_1. (!e b x. happ (happ (happ _1 e) b) x = happ (happ _2 e) b) ==> (!_0. (!e b. happ (happ _0 e) b = happ (happ _1 e) b) ==> (!e. happ h4/sum/INR e = h4/sum/ABS__sum (happ _0 e))))
% Assm [h4s_sums_sumu_u_ISOu_u_DEFu_c1]: !r. h4/sum/IS__SUM__REP r <=> h4/sum/REP__sum (h4/sum/ABS__sum r) = r
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_sums_INRu_u_11]: !y x. happ h4/sum/INR x = happ h4/sum/INR y <=> x = y
% Assm [h4s_bools_SELECTu_u_REFLu_u_2]: !_0. (!x y. happ (happ _0 x) y <=> x = y) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_sums_ISu_u_SUMu_u_REP0]: !f. h4/sum/IS__SUM__REP f <=> (?v1 v2. (!x x x. happ (happ (happ f x) x) x <=> x = v1 /\ x) \/ (!x x x. happ (happ (happ f x) x) x <=> x = v2 /\ ~x))
% 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_predu_u_sets_IMAGEu_u_DEF]: !_0. (!f s x. happ (happ (happ _0 f) s) x = h4/pair/_2C (happ f x) (h4/bool/IN x s)) ==> (!s f. h4/pred__set/IMAGE f s = h4/pred__set/GSPEC (happ (happ _0 f) s))
% Assm [h4s_combins_ou_u_ABSu_u_R]: !_0. (!g x. happ (happ _0 g) x = happ g x) ==> (!g f x. happ (happ (happ h4/combin/o f) (happ _0 g)) x = happ f (happ g x))
% Assm [h4s_combins_GENu_u_literalu_u_caseu_u_RAND]: !v f P. happ P (h4/bool/literal__case f v) = h4/bool/literal__case (happ (happ h4/combin/o P) f) v
% Assm [h4s_combins_GENu_u_LETu_u_RAND]: !v f P. happ P (h4/bool/LET f v) = h4/bool/LET (happ (happ h4/combin/o P) f) v
% Assm [h4s_combins_ou_u_ASSOC]: !h g f. happ (happ h4/combin/o f) (happ (happ h4/combin/o g) h) = happ (happ h4/combin/o (happ (happ h4/combin/o f) g)) h
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_COMPOSE]: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (happ (happ h4/combin/o f) g) x
% Assm [h4s_combins_Ku_u_ou_u_THMu_c0]: !v f. happ (happ h4/combin/o (h4/combin/K v)) f = h4/combin/K v
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_predu_u_sets_INJu_u_IFF]: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> (happ f x = happ f y <=> x = y))
% Assm [h4s_whiles_HOAREu_u_SPECu_u_DEF]: !Q P C. h4/while/HOARE__SPEC P C Q <=> (!s. happ P s ==> happ Q (happ C s))
% Assm [h4s_predu_u_sets_INJu_u_INSERT]: !x t s f. h4/pred__set/INJ f (h4/pred__set/INSERT x s) t <=> h4/pred__set/INJ f s t /\ h4/bool/IN (happ f x) t /\ (!y. h4/bool/IN y s /\ happ f x = happ f y ==> x = y)
% Assm [h4s_predu_u_sets_INJu_u_DELETE]: !t s f. h4/pred__set/INJ f s t ==> (!e. h4/bool/IN e s ==> h4/pred__set/INJ f (h4/pred__set/DELETE s e) (h4/pred__set/DELETE t (happ f e)))
% Assm [h4s_predu_u_sets_IMAGEu_u_IN]: !x s. h4/bool/IN x s ==> (!f. h4/bool/IN (happ f x) (h4/pred__set/IMAGE f s))
% Assm [h4s_predu_u_sets_IMAGEu_u_INSERT]: !x s f. h4/pred__set/IMAGE f (h4/pred__set/INSERT x s) = h4/pred__set/INSERT (happ f x) (h4/pred__set/IMAGE f s)
% Assm [h4s_predu_u_sets_IMAGEu_u_11]: !s2 s1 f. (!x y. happ f x = happ f y <=> x = y) ==> (h4/pred__set/IMAGE f s1 = h4/pred__set/IMAGE f s2 <=> s1 = s2)
% Assm [h4s_predu_u_sets_IMAGEu_u_CONG]: !s_27 s f_27 f. s = s_27 /\ (!x. h4/bool/IN x s_27 ==> happ f x = happ f_27 x) ==> h4/pred__set/IMAGE f s = h4/pred__set/IMAGE f_27 s_27
% Assm [h4s_predu_u_sets_IMAGEu_u_DELETE]: !x s f. ~h4/bool/IN x s ==> h4/pred__set/IMAGE f (h4/pred__set/DELETE s x) = h4/pred__set/IMAGE f s
% Assm [h4s_bools_literalu_u_caseu_u_THM]: !x f. h4/bool/literal__case f x = happ f x
% Assm [h4s_predu_u_sets_IMAGEu_u_INTER]: !t s f. h4/pred__set/SUBSET (h4/pred__set/IMAGE f (h4/pred__set/INTER s t)) (h4/pred__set/INTER (h4/pred__set/IMAGE f s) (h4/pred__set/IMAGE f t))
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_relations_TFLu_u_INDUCTIVEu_u_INVARIANTu_u_ONu_u_WFREC]: !x f R P M D. f = h4/relation/WFREC R M /\ h4/relation/WF R /\ h4/relation/INDUCTIVE__INVARIANT__ON R D P M /\ happ D x ==> happ (happ P x) (happ f x)
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_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_options_OPTIONu_u_APPLYu_u_defu_c1]: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c0]: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_combins_Ku_u_THM]: !y x. happ (h4/combin/K x) y = x
% 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_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_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_INu_u_DELETE]: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. happ P x ==> Q) <=> (?x. happ P x) ==> Q
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_predu_u_sets_DIFFu_u_INSERT]: !x t s. h4/pred__set/DIFF s (h4/pred__set/INSERT x t) = h4/pred__set/DIFF (h4/pred__set/DELETE s x) t
% Assm [h4s_predu_u_sets_DELETEu_u_DEF]: !x s. h4/pred__set/DELETE s x = h4/pred__set/DIFF s (h4/pred__set/INSERT x h4/pred__set/EMPTY)
% Assm [h4s_predu_u_sets_INu_u_DIFF]: !x t s. h4/bool/IN x (h4/pred__set/DIFF s t) <=> h4/bool/IN x s /\ ~h4/bool/IN x t
% Assm [h4s_predu_u_sets_DIFFu_u_EMPTY]: !s. h4/pred__set/DIFF s h4/pred__set/EMPTY = s
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_relations_INDUCTIVEu_u_INVARIANTu_u_ONu_u_WFREC]: !x R P M D. h4/relation/WF R /\ h4/relation/INDUCTIVE__INVARIANT__ON R D P M /\ happ D x ==> happ (happ P x) (happ (h4/relation/WFREC R M) x)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Goal: !u t s g f. h4/pred__set/SURJ f s t /\ h4/pred__set/SURJ g t u ==> h4/pred__set/SURJ (happ (happ h4/combin/o g) f) s u
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1238025,TV_Q1238021]: ![V_f, V_g]: (![V_x]: s(TV_Q1238021,happ(s(t_fun(TV_Q1238025,TV_Q1238021),V_f),s(TV_Q1238025,V_x))) = s(TV_Q1238021,happ(s(t_fun(TV_Q1238025,TV_Q1238021),V_g),s(TV_Q1238025,V_x))) => s(t_fun(TV_Q1238025,TV_Q1238021),V_f) = s(t_fun(TV_Q1238025,TV_Q1238021),V_g))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_predu_u_sets_INJu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t))))) & ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s))))) => (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_predu_u_sets_INJu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_u, V_t, V_s, V_g, V_f]: ((p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27b,t_bool),V_t),s(t_fun(TV_u_27c,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o),s(t_fun(TV_u_27b,TV_u_27c),V_g))),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_27c,t_bool),V_u)))))).
fof(ah4s_predu_u_sets_SURJu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_bool),V_t)))) => ?[V_y]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,V_x)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
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,t0)))).
fof(ah4s_predu_u_sets_IMAGEu_u_COMPOSE, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_s, V_g, V_f]: s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o),s(t_fun(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),V_s))) = s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_SURJu_u_ID, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_uu_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x) => ![V_s]: p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27a,TV_u_27a),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_sums_sumu_u_axiom, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_f, V_g, V_h]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g))),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h)))) <=> (s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h))),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl))) = s(t_fun(TV_u_27a,TV_u_27c),V_f) & s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_fun(TV_u_27b,TV_u_27c))),h4s_combins_o),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h))),s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr))) = s(t_fun(TV_u_27b,TV_u_27c),V_g))) => ![V_g, V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g)))))))).
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_sums_sumu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f1u_27, V_f1, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_M) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) & (![V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))) => s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_fu_27),s(TV_u_27a,V_x)))) & ![V_y]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))) => s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1u_27),s(TV_u_27b,V_y)))))) => s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_M),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27),s(t_fun(TV_u_27a,TV_u_27c),V_fu_27),s(t_fun(TV_u_27b,TV_u_27c),V_f1u_27))))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y)))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_sums_sumu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_ss]: (?[V_x]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))) | ?[V_y]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))))).
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_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_o, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27c)),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)))),h4s_options_some(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o))),s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27b)),V_g))),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27b)),V_g),s(t_h4s_options_option(TV_u_27a),V_x)))))).
fof(ah4s_sums_INLu_u_11, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
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_sums_INLu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_2]: (![V_x, V_e, V_b, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool)))),V_uu_2),s(TV_u_27a,V_x))),s(TV_u_27a,V_e))),s(t_bool,V_b))),s(TV_u_27b,V_y)))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_e) & p(s(t_bool,V_b)))) => ![V_uu_1]: (![V_e, V_b, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_e))),s(t_bool,V_b))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool)))),V_uu_2),s(TV_u_27a,V_x))),s(TV_u_27a,V_e))),s(t_bool,V_b))) => ![V_uu_0]: (![V_e, V_b]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27a,V_e))),s(t_bool,V_b))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_e))),s(t_bool,V_b))) => ![V_e]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_e))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27a,V_e))))))))).
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_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
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,t0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_sums_INRu_u_nequ_u_INL, axiom, ![TV_u_27b,TV_u_27a]: ![V_v2, V_v1]: ~ (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_v2))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_v1))))).
fof(ah4s_sums_sumu_u_ISOu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_a]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),h4s_sums_repu_u_sum(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_a))))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_a)).
fof(ah4s_sums_INRu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_e, V_b, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),V_uu_2),s(TV_u_27b,V_e))),s(t_bool,V_b))),s(TV_u_27b,V_y)))) <=> (s(TV_u_27b,V_y) = s(TV_u_27b,V_e) & ~ (p(s(t_bool,V_b))))) => ![V_uu_1]: (![V_e, V_b, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27b,V_e))),s(t_bool,V_b))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),V_uu_2),s(TV_u_27b,V_e))),s(t_bool,V_b))) => ![V_uu_0]: (![V_e, V_b]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27b,V_e))),s(t_bool,V_b))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27b,V_e))),s(t_bool,V_b))) => ![V_e]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_e))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27b,V_e))))))))).
fof(ah4s_sums_sumu_u_ISOu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_r]: (p(s(t_bool,h4s_sums_isu_u_sumu_u_rep(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r)))) <=> s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),h4s_sums_repu_u_sum(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r))))) = s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r))).
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_sums_INRu_u_11, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_y))).
fof(ah4s_bools_SELECTu_u_REFLu_u_2, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ah4s_sums_ISu_u_SUMu_u_REP0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: (p(s(t_bool,h4s_sums_isu_u_sumu_u_rep(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f)))) <=> ?[V_v1, V_v2]: (![V_x, V_x0, V_x1]: (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)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f),s(t_bool,V_x))),s(TV_u_27a,V_x0))),s(TV_u_27b,V_x1)))) <=> (s(TV_u_27a,V_x0) = s(TV_u_27a,V_v1) & p(s(t_bool,V_x)))) | ![V_x, V_x0, V_x1]: (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)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f),s(t_bool,V_x))),s(TV_u_27a,V_x0))),s(TV_u_27b,V_x1)))) <=> (s(TV_u_27b,V_x1) = s(TV_u_27b,V_v2) & ~ (p(s(t_bool,V_x)))))))).
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_predu_u_sets_IMAGEu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_s, V_x]: s(t_h4s_pairs_prod(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27b,t_bool),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_s, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_combins_ou_u_ABSu_u_R, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_0]: (![V_g, V_x]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) => ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_combins_GENu_u_literalu_u_caseu_u_RAND, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_v, V_f, V_P]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,V_v))))) = s(TV_u_27a,h4s_bools_literalu_u_case(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a))),h4s_combins_o),s(t_fun(TV_u_27b,TV_u_27a),V_P))),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(TV_u_27c,V_v)))).
fof(ah4s_combins_GENu_u_LETu_u_RAND, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_v, V_f, V_P]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,V_v))))) = s(TV_u_27a,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a))),h4s_combins_o),s(t_fun(TV_u_27b,TV_u_27a),V_P))),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(TV_u_27c,V_v)))).
fof(ah4s_combins_ou_u_ASSOC, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27d,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27a),t_fun(TV_u_27d,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27a),t_fun(TV_u_27d,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27d,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,TV_u_27a))),h4s_combins_o),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(t_fun(TV_u_27d,TV_u_27c),V_h))))) = s(t_fun(TV_u_27d,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))))),s(t_fun(TV_u_27d,TV_u_27c),V_h)))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),h4s_combins_o),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(t_h4s_options_option(TV_u_27a),V_x)))).
fof(ah4s_combins_Ku_u_ou_u_THMu_c0, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_v, V_f]: s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o),s(t_fun(TV_u_27b,TV_u_27c),h4s_combins_k(s(TV_u_27c,V_v))))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_k(s(TV_u_27c,V_v)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_predu_u_sets_INJu_u_IFF, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t))))) & ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s))))) => (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_whiles_HOAREu_u_SPECu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_Q, V_P, V_C]: (p(s(t_bool,h4s_whiles_hoareu_u_spec(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,TV_u_27b),V_C),s(t_fun(TV_u_27b,t_bool),V_Q)))) <=> ![V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_Q),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_C),s(TV_u_27a,V_s))))))))).
fof(ah4s_predu_u_sets_INJu_u_INSERT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & (p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t)))) & ![V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_predu_u_sets_INJu_u_DELETE, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) => ![V_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_e))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27b,t_bool),V_t),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_e))))))))))).
fof(ah4s_predu_u_sets_IMAGEu_u_IN, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f]: p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_IMAGEu_u_INSERT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_s, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_IMAGEu_u_11, axiom, ![TV_u_27b,TV_u_27a]: ![V_s2, V_s1, V_f]: (![V_x, V_y]: (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => (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_s1))) = 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_s2))) <=> s(t_fun(TV_u_27a,t_bool),V_s1) = s(t_fun(TV_u_27a,t_bool),V_s2)))).
fof(ah4s_predu_u_sets_IMAGEu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_su_27, V_s, V_fu_27, V_f]: ((s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_su_27) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_su_27)))) => 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(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))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_fu_27),s(t_fun(TV_u_27a,t_bool),V_su_27))))).
fof(ah4s_predu_u_sets_IMAGEu_u_DELETE, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_s, V_f]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) => s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_bools_literalu_u_caseu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_predu_u_sets_IMAGEu_u_INTER, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_inter(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))),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_t))))))))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_relations_TFLu_u_INDUCTIVEu_u_INVARIANTu_u_ONu_u_WFREC, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f, V_R, V_P, V_M, V_D]: ((s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) & (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & (p(s(t_bool,h4s_relations_inductiveu_u_invariantu_u_on(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_D),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_D),s(TV_u_27a,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_predu_u_sets_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_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_options_OPTIONu_u_APPLYu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_some(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(t_h4s_options_option(TV_u_27b),V_x)))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_none),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
fof(ah4s_sats_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_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_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),h4s_combins_k(s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
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_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t0) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t0))) = 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_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_INu_u_DELETE, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_FORALLu_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,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t0))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))))).
fof(ah4s_bools_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_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
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_predu_u_sets_DIFFu_u_INSERT, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),V_t)))).
fof(ah4s_predu_u_sets_DELETEu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),V_s),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_DIFF, 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_diff(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_DIFFu_u_EMPTY, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t0))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_relations_INDUCTIVEu_u_INVARIANTu_u_ONu_u_WFREC, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_R, V_P, V_M, V_D]: ((p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & (p(s(t_bool,h4s_relations_inductiveu_u_invariantu_u_on(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_D),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_D),s(TV_u_27a,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),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f0))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ch4s_predu_u_sets_SURJu_u_COMPOSE, conjecture, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_u, V_t, V_s, V_g, V_f]: ((p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27b,t_bool),V_t),s(t_fun(TV_u_27c,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o),s(t_fun(TV_u_27b,TV_u_27c),V_g))),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_27c,t_bool),V_u)))))).
