%   ORIGINAL: h4/set__relation/tc__strongind__right
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/set__relation/tc__def: h4/set__relation/tc = (\r a0. !tc_27. (!a00. (?x y. a00 = h4/pair/_2C x y /\ r (h4/pair/_2C x y)) \/ (?x y. a00 = h4/pair/_2C x y /\ (?z. tc_27 (h4/pair/_2C x z) /\ tc_27 (h4/pair/_2C z y))) ==> tc_27 a00) ==> tc_27 a0)
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/set__relation/tc__ind: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> tc_27 x y) /\ (!x y. (?z. tc_27 x z /\ tc_27 z y) ==> tc_27 x y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> tc_27 x y)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/set__relation/tc__rules_c0: !y x r. h4/bool/IN (h4/pair/_2C x y) r ==> h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r)
% Assm: h4/set__relation/tc__ind__right: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> tc_27 x y) /\ (!x y. (?z. tc_27 x z /\ h4/bool/IN (h4/pair/_2C z y) r) ==> tc_27 x y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> tc_27 x y)
% Assm: h4/set__relation/tc__strongind: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> tc_27 x y) /\ (!x y. (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ tc_27 x z /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r) /\ tc_27 z y) ==> tc_27 x y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> tc_27 x y)
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/set__relation/tc__rules_c1: !y x r. (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r)) ==> h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r)
% Assm: h4/set__relation/tc__ind__left: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> tc_27 x y) /\ (!x y. (?z. h4/bool/IN (h4/pair/_2C x z) r /\ tc_27 z y) ==> tc_27 x y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> tc_27 x y)
% Assm: h4/set__relation/rtc__ind__right: !tc_27 r. (!x. h4/bool/IN x (h4/set__relation/domain r) \/ h4/bool/IN x (h4/set__relation/range r) ==> tc_27 x x) /\ (!x y. (?z. tc_27 x z /\ h4/bool/IN (h4/pair/_2C z y) r) ==> tc_27 x y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> tc_27 x y)
% Assm: h4/set__relation/tc__strongind__left: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> tc_27 x y) /\ (!x y. (?z. h4/bool/IN (h4/pair/_2C x z) r /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r) /\ tc_27 z y) ==> tc_27 x y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> tc_27 x y)
% Assm: h4/set__relation/tc__cases__right: !y x r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) <=> h4/bool/IN (h4/pair/_2C x y) r \/ (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ h4/bool/IN (h4/pair/_2C z y) r)
% Assm: h4/set__relation/tc__cases: !y x r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) <=> h4/bool/IN (h4/pair/_2C x y) r \/ (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r))
% Assm: h4/set__relation/tc__cases__left: !y x r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) <=> h4/bool/IN (h4/pair/_2C x y) r \/ (?z. h4/bool/IN (h4/pair/_2C x z) r /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r))
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/set__relation/tc__strongind0: !tc_27 r. (!x y. r (h4/pair/_2C x y) ==> tc_27 (h4/pair/_2C x y)) /\ (!x y. (?z. h4/set__relation/tc r (h4/pair/_2C x z) /\ tc_27 (h4/pair/_2C x z) /\ h4/set__relation/tc r (h4/pair/_2C z y) /\ tc_27 (h4/pair/_2C z y)) ==> tc_27 (h4/pair/_2C x y)) ==> (!a0. h4/set__relation/tc r a0 ==> tc_27 a0)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/set__relation/rrestrict__def: !s r. h4/set__relation/rrestrict r s = h4/pred__set/GSPEC (h4/pair/UNCURRY (\x y. h4/pair/_2C (h4/pair/_2C x y) (h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN x s /\ h4/bool/IN y s)))
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/pair/LAMBDA__PROD: !P. (\p. P p) = h4/pair/UNCURRY (\p1 p2. P (h4/pair/_2C p1 p2))
% Assm: h4/pair/PFORALL__THM: !P. (!x y. P x y) <=> $forall (h4/pair/UNCURRY (\x y. P x y))
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/set__relation/strict__def: !r. h4/set__relation/strict r = h4/pred__set/GSPEC (h4/pair/UNCURRY (\x y. h4/pair/_2C (h4/pair/_2C x y) (h4/bool/IN (h4/pair/_2C x y) r /\ ~(x = y))))
% Assm: h4/set__relation/in__rrestrict: !y x s r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/rrestrict r s) <=> h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN x s /\ h4/bool/IN y s
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/set__relation/tc__rules0_c1: !y x r. (?z. h4/set__relation/tc r (h4/pair/_2C x z) /\ h4/set__relation/tc r (h4/pair/_2C z y)) ==> h4/set__relation/tc r (h4/pair/_2C x y)
% Assm: h4/set__relation/tc__cases0: !r a0. h4/set__relation/tc r a0 <=> (?x y. a0 = h4/pair/_2C x y /\ r (h4/pair/_2C x y)) \/ (?x y. a0 = h4/pair/_2C x y /\ (?z. h4/set__relation/tc r (h4/pair/_2C x z) /\ h4/set__relation/tc r (h4/pair/_2C z y)))
% Assm: h4/set__relation/tc__rules0_c0: !y x r. r (h4/pair/_2C x y) ==> h4/set__relation/tc r (h4/pair/_2C x y)
% Assm: h4/set__relation/tc__ind0: !tc_27 r. (!x y. r (h4/pair/_2C x y) ==> tc_27 (h4/pair/_2C x y)) /\ (!x y. (?z. tc_27 (h4/pair/_2C x z) /\ tc_27 (h4/pair/_2C z y)) ==> tc_27 (h4/pair/_2C x y)) ==> (!a0. h4/set__relation/tc r a0 ==> tc_27 a0)
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% 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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/set__relation/univ__reln__def: !xs. h4/set__relation/univ__reln xs = h4/pred__set/GSPEC (h4/pair/UNCURRY (\x1 x2. h4/pair/_2C (h4/pair/_2C x1 x2) (h4/bool/IN x1 xs /\ h4/bool/IN x2 xs)))
% Assm: h4/set__relation/domain__def: !r. h4/set__relation/domain r = h4/pred__set/GSPEC (\x. h4/pair/_2C x (?y. h4/bool/IN (h4/pair/_2C x y) r))
% Assm: h4/set__relation/range__def: !r. h4/set__relation/range r = h4/pred__set/GSPEC (\y. h4/pair/_2C y (?x. h4/bool/IN (h4/pair/_2C x y) r))
% Assm: h4/set__relation/finite__prefixes__subset: !s_27 s r. h4/set__relation/finite__prefixes r s /\ h4/pred__set/SUBSET s_27 s ==> h4/set__relation/finite__prefixes r s_27 /\ h4/set__relation/finite__prefixes (h4/set__relation/rrestrict r s_27) s_27
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/set__relation/rrestrict__rrestrict: !y x r. h4/set__relation/rrestrict (h4/set__relation/rrestrict r x) y = h4/set__relation/rrestrict r (h4/pred__set/INTER x y)
% Assm: h4/set__relation/rrestrict__union: !s r2 r1. h4/set__relation/rrestrict (h4/pred__set/UNION r1 r2) s = h4/pred__set/UNION (h4/set__relation/rrestrict r1 s) (h4/set__relation/rrestrict r2 s)
% Assm: h4/set__relation/strict__rrestrict: !s r. h4/set__relation/strict (h4/set__relation/rrestrict r s) = h4/set__relation/rrestrict (h4/set__relation/strict r) s
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pair/LEX__DEF: !R2 R1. h4/pair/LEX R1 R2 = h4/pair/UNCURRY (\s t. h4/pair/UNCURRY (\u v. R1 s u \/ s = u /\ R2 t v))
% Assm: h4/pred__set/partition__def: !s R. h4/pred__set/partition R s = h4/pred__set/GSPEC (\t. h4/pair/_2C t (?x. h4/bool/IN x s /\ t = h4/pred__set/GSPEC (\y. h4/pair/_2C y (h4/bool/IN y s /\ R x y))))
% Assm: h4/pred__set/KoenigsLemma: !R. (!x. h4/pred__set/FINITE (h4/pred__set/GSPEC (\y. h4/pair/_2C y (R x y)))) ==> (!x. ~h4/pred__set/FINITE (h4/pred__set/GSPEC (\y. h4/pair/_2C y (h4/relation/RTC R x y))) ==> (?f. f h4/num/0 = x /\ (!n. R (f n) (f (h4/num/SUC n)))))
% 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/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/pred__set/KoenigsLemma__WF: !R. (!x. h4/pred__set/FINITE (h4/pred__set/GSPEC (\y. h4/pair/_2C y (R x y)))) /\ h4/relation/WF (h4/relation/inv R) ==> (!x. h4/pred__set/FINITE (h4/pred__set/GSPEC (\y. h4/pair/_2C y (h4/relation/RTC R x y))))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/pair/LEX__DEF__THM: !d c b a R2 R1. h4/pair/LEX R1 R2 (h4/pair/_2C a b) (h4/pair/_2C c d) <=> R1 a c \/ a = c /\ R2 b d
% Assm: h4/set__relation/finite__prefixes__def: !s r. h4/set__relation/finite__prefixes r s <=> (!e. h4/bool/IN e s ==> h4/pred__set/FINITE (h4/pred__set/GSPEC (\e_27. h4/pair/_2C e_27 (h4/bool/IN (h4/pair/_2C e_27 e) r))))
% Assm: h4/pred__set/GSPEC__AND: !Q P. h4/pred__set/GSPEC (\x. h4/pair/_2C x (P x /\ Q x)) = h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (P x))) (h4/pred__set/GSPEC (\x. h4/pair/_2C x (Q x)))
% Assm: h4/pred__set/GSPEC__ID: !y. h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN x y)) = y
% Assm: h4/pred__set/INTER__FINITE: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/FINITE (h4/pred__set/INTER s t))
% 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__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/prim__rec/WF__IFF__WELLFOUNDED: !R. h4/relation/WF R <=> h4/prim__rec/wellfounded R
% Assm: h4/prim__rec/wellfounded__def: !R. h4/prim__rec/wellfounded R <=> ~(?f. !n. R (f (h4/num/SUC n)) (f n))
% Assm: h4/relation/inv__DEF: !y x R. h4/relation/inv R x y <=> R y x
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/pred__set/IN__BIGUNION: !x sos. h4/bool/IN x (h4/pred__set/BIGUNION sos) <=> (?s. h4/bool/IN x s /\ h4/bool/IN s sos)
% Assm: h4/pred__set/FINITE__BIGUNION__EQ: !P. h4/pred__set/FINITE (h4/pred__set/BIGUNION P) <=> h4/pred__set/FINITE P /\ (!s. h4/bool/IN s P ==> h4/pred__set/FINITE s)
% Assm: h4/pred__set/IMAGE__FINITE: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm: h4/pred__set/FINITE__INSERT: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% 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/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/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/relation/RTC__CASES1: !y x R. h4/relation/RTC R x y <=> x = y \/ (?u. R x u /\ h4/relation/RTC R u y)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/FALSITY: !t. F ==> t
% 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)
% Goal: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> tc_27 x y) /\ (!x y. (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ tc_27 x z /\ h4/bool/IN (h4/pair/_2C z y) r) ==> tc_27 x y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> tc_27 x y)
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_setu_u_relations_tcu_u_def]: !x x. happ (h4/set__relation/tc x) x <=> (!tc_27. (!a00. (?x y. a00 = h4/pair/_2C x y /\ happ x (h4/pair/_2C x y)) \/ (?x y. a00 = h4/pair/_2C x y /\ (?z. happ tc_27 (h4/pair/_2C x z) /\ happ tc_27 (h4/pair/_2C z y))) ==> happ tc_27 a00) ==> happ tc_27 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_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_setu_u_relations_tcu_u_ind]: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> happ (happ tc_27 x) y) /\ (!x y. (?z. happ (happ tc_27 x) z /\ happ (happ tc_27 z) y) ==> happ (happ tc_27 x) y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> happ (happ tc_27 x) y)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_setu_u_relations_tcu_u_rulesu_c0]: !y x r. h4/bool/IN (h4/pair/_2C x y) r ==> h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r)
% Assm [h4s_setu_u_relations_tcu_u_indu_u_right]: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> happ (happ tc_27 x) y) /\ (!x y. (?z. happ (happ tc_27 x) z /\ h4/bool/IN (h4/pair/_2C z y) r) ==> happ (happ tc_27 x) y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> happ (happ tc_27 x) y)
% Assm [h4s_setu_u_relations_tcu_u_strongind]: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> happ (happ tc_27 x) y) /\ (!x y. (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ happ (happ tc_27 x) z /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r) /\ happ (happ tc_27 z) y) ==> happ (happ tc_27 x) y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> happ (happ tc_27 x) y)
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_setu_u_relations_tcu_u_rulesu_c1]: !y x r. (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r)) ==> h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r)
% Assm [h4s_setu_u_relations_tcu_u_indu_u_left]: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> happ (happ tc_27 x) y) /\ (!x y. (?z. h4/bool/IN (h4/pair/_2C x z) r /\ happ (happ tc_27 z) y) ==> happ (happ tc_27 x) y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> happ (happ tc_27 x) y)
% Assm [h4s_setu_u_relations_rtcu_u_indu_u_right]: !tc_27 r. (!x. h4/bool/IN x (h4/set__relation/domain r) \/ h4/bool/IN x (h4/set__relation/range r) ==> happ (happ tc_27 x) x) /\ (!x y. (?z. happ (happ tc_27 x) z /\ h4/bool/IN (h4/pair/_2C z y) r) ==> happ (happ tc_27 x) y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> happ (happ tc_27 x) y)
% Assm [h4s_setu_u_relations_tcu_u_strongindu_u_left]: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> happ (happ tc_27 x) y) /\ (!x y. (?z. h4/bool/IN (h4/pair/_2C x z) r /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r) /\ happ (happ tc_27 z) y) ==> happ (happ tc_27 x) y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> happ (happ tc_27 x) y)
% Assm [h4s_setu_u_relations_tcu_u_casesu_u_right]: !y x r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) <=> h4/bool/IN (h4/pair/_2C x y) r \/ (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ h4/bool/IN (h4/pair/_2C z y) r)
% Assm [h4s_setu_u_relations_tcu_u_cases]: !y x r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) <=> h4/bool/IN (h4/pair/_2C x y) r \/ (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r))
% Assm [h4s_setu_u_relations_tcu_u_casesu_u_left]: !y x r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) <=> h4/bool/IN (h4/pair/_2C x y) r \/ (?z. h4/bool/IN (h4/pair/_2C x z) r /\ h4/bool/IN (h4/pair/_2C z y) (h4/set__relation/tc r))
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_setu_u_relations_tcu_u_strongind0]: !tc_27 r. (!x y. happ r (h4/pair/_2C x y) ==> happ tc_27 (h4/pair/_2C x y)) /\ (!x y. (?z. happ (h4/set__relation/tc r) (h4/pair/_2C x z) /\ happ tc_27 (h4/pair/_2C x z) /\ happ (h4/set__relation/tc r) (h4/pair/_2C z y) /\ happ tc_27 (h4/pair/_2C z y)) ==> happ tc_27 (h4/pair/_2C x y)) ==> (!a0. happ (h4/set__relation/tc r) a0 ==> happ tc_27 a0)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_setu_u_relations_rrestrictu_u_def]: !_1. (!r x s y. ?v. (v <=> h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN x s /\ h4/bool/IN y s) /\ happ (happ (happ (happ _1 r) x) s) y = h4/pair/_2C (h4/pair/_2C x y) v) ==> (!_0. (!r s x. happ (happ (happ _0 r) s) x = happ (happ (happ _1 r) x) s) ==> (!s r. h4/set__relation/rrestrict r s = h4/pred__set/GSPEC (h4/pair/UNCURRY (happ (happ _0 r) s))))
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_pairs_LAMBDAu_u_PROD]: !_1. (!P p1 p2. happ (happ (happ _1 P) p1) p2 = happ P (h4/pair/_2C p1 p2)) ==> (!_0. (!P p1. happ (happ _0 P) p1 = happ (happ _1 P) p1) ==> (!P x. happ P x = happ (h4/pair/UNCURRY (happ _0 P)) x))
% Assm [h4s_pairs_PFORALLu_u_THM]: !_1. (!P x y. happ (happ (happ _1 P) x) y <=> happ (happ P x) y) ==> (!_0. (!P x. happ (happ _0 P) x = happ (happ _1 P) x) ==> (!P. (!x y. happ (happ P x) y) <=> $forall (h4/pair/UNCURRY (happ _0 P))))
% 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_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_setu_u_relations_strictu_u_def]: !_1. (!r x y. ?v. (v <=> h4/bool/IN (h4/pair/_2C x y) r /\ ~(x = y)) /\ happ (happ (happ _1 r) x) y = h4/pair/_2C (h4/pair/_2C x y) v) ==> (!_0. (!r x. happ (happ _0 r) x = happ (happ _1 r) x) ==> (!r. h4/set__relation/strict r = h4/pred__set/GSPEC (h4/pair/UNCURRY (happ _0 r))))
% Assm [h4s_setu_u_relations_inu_u_rrestrict]: !y x s r. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/rrestrict r s) <=> h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN x s /\ h4/bool/IN y s
% 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_setu_u_relations_tcu_u_rules0u_c1]: !y x r. (?z. happ (h4/set__relation/tc r) (h4/pair/_2C x z) /\ happ (h4/set__relation/tc r) (h4/pair/_2C z y)) ==> happ (h4/set__relation/tc r) (h4/pair/_2C x y)
% Assm [h4s_setu_u_relations_tcu_u_cases0]: !r a0. happ (h4/set__relation/tc r) a0 <=> (?x y. a0 = h4/pair/_2C x y /\ happ r (h4/pair/_2C x y)) \/ (?x y. a0 = h4/pair/_2C x y /\ (?z. happ (h4/set__relation/tc r) (h4/pair/_2C x z) /\ happ (h4/set__relation/tc r) (h4/pair/_2C z y)))
% Assm [h4s_setu_u_relations_tcu_u_rules0u_c0]: !y x r. happ r (h4/pair/_2C x y) ==> happ (h4/set__relation/tc r) (h4/pair/_2C x y)
% Assm [h4s_setu_u_relations_tcu_u_ind0]: !tc_27 r. (!x y. happ r (h4/pair/_2C x y) ==> happ tc_27 (h4/pair/_2C x y)) /\ (!x y. (?z. happ tc_27 (h4/pair/_2C x z) /\ happ tc_27 (h4/pair/_2C z y)) ==> happ tc_27 (h4/pair/_2C x y)) ==> (!a0. happ (h4/set__relation/tc r) a0 ==> happ tc_27 a0)
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. happ P x ==> Q) <=> (?x. happ P x) ==> Q
% 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_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_setu_u_relations_univu_u_relnu_u_def]: !_1. (!x1 xs x2. ?v. (v <=> h4/bool/IN x1 xs /\ h4/bool/IN x2 xs) /\ happ (happ (happ _1 x1) xs) x2 = h4/pair/_2C (h4/pair/_2C x1 x2) v) ==> (!_0. (!xs x1. happ (happ _0 xs) x1 = happ (happ _1 x1) xs) ==> (!xs. h4/set__relation/univ__reln xs = h4/pred__set/GSPEC (h4/pair/UNCURRY (happ _0 xs))))
% Assm [h4s_setu_u_relations_domainu_u_def]: !_0. (!r x. ?v. (v <=> (?y. h4/bool/IN (h4/pair/_2C x y) r)) /\ happ (happ _0 r) x = h4/pair/_2C x v) ==> (!r. h4/set__relation/domain r = h4/pred__set/GSPEC (happ _0 r))
% Assm [h4s_setu_u_relations_rangeu_u_def]: !_0. (!r y. ?v. (v <=> (?x. h4/bool/IN (h4/pair/_2C x y) r)) /\ happ (happ _0 r) y = h4/pair/_2C y v) ==> (!r. h4/set__relation/range r = h4/pred__set/GSPEC (happ _0 r))
% Assm [h4s_setu_u_relations_finiteu_u_prefixesu_u_subset]: !s_27 s r. h4/set__relation/finite__prefixes r s /\ h4/pred__set/SUBSET s_27 s ==> h4/set__relation/finite__prefixes r s_27 /\ h4/set__relation/finite__prefixes (h4/set__relation/rrestrict r s_27) s_27
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_setu_u_relations_rrestrictu_u_rrestrict]: !y x r. h4/set__relation/rrestrict (h4/set__relation/rrestrict r x) y = h4/set__relation/rrestrict r (h4/pred__set/INTER x y)
% Assm [h4s_setu_u_relations_rrestrictu_u_union]: !s r2 r1. h4/set__relation/rrestrict (h4/pred__set/UNION r1 r2) s = h4/pred__set/UNION (h4/set__relation/rrestrict r1 s) (h4/set__relation/rrestrict r2 s)
% Assm [h4s_setu_u_relations_strictu_u_rrestrict]: !s r. h4/set__relation/strict (h4/set__relation/rrestrict r s) = h4/set__relation/rrestrict (h4/set__relation/strict r) s
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_pairs_LEXu_u_DEF]: !_3. (!R1 s u R2 t v. happ (happ (happ (happ (happ (happ _3 R1) s) u) R2) t) v <=> happ (happ R1 s) u \/ s = u /\ happ (happ R2 t) v) ==> (!_2. (!R1 s R2 t u. happ (happ (happ (happ (happ _2 R1) s) R2) t) u = happ (happ (happ (happ (happ _3 R1) s) u) R2) t) ==> (!_1. (!R1 s R2 t. happ (happ (happ (happ _1 R1) s) R2) t = h4/pair/UNCURRY (happ (happ (happ (happ _2 R1) s) R2) t)) ==> (!_0. (!R1 R2 s. happ (happ (happ _0 R1) R2) s = happ (happ (happ _1 R1) s) R2) ==> (!R2 R1. h4/pair/LEX R1 R2 = h4/pair/UNCURRY (happ (happ _0 R1) R2)))))
% Assm [h4s_predu_u_sets_partitionu_u_def]: !_1. (!s R x y. ?v. (v <=> h4/bool/IN y s /\ happ (happ R x) y) /\ happ (happ (happ (happ _1 s) R) x) y = h4/pair/_2C y v) ==> (!_0. (!s R t. ?v. (v <=> (?x. h4/bool/IN x s /\ t = h4/pred__set/GSPEC (happ (happ (happ _1 s) R) x))) /\ happ (happ (happ _0 s) R) t = h4/pair/_2C t v) ==> (!s R. h4/pred__set/partition R s = h4/pred__set/GSPEC (happ (happ _0 s) R)))
% Assm [h4s_predu_u_sets_KoenigsLemma]: !_0. (!R x y. happ (happ (happ _0 R) x) y = h4/pair/_2C y (happ (happ R x) y)) ==> (!R. (!x. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 R) x))) ==> (!x. ~h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 (h4/relation/RTC R)) x)) ==> (?f. happ f h4/num/0 = x /\ (!n. happ (happ R (happ f n)) (happ f (h4/num/SUC n))))))
% 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_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_predu_u_sets_KoenigsLemmau_u_WF]: !_0. (!R x y. happ (happ (happ _0 R) x) y = h4/pair/_2C y (happ (happ R x) y)) ==> (!R. (!x. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 R) x))) /\ h4/relation/WF (h4/relation/inv R) ==> (!x. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 (h4/relation/RTC R)) x))))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_pairs_LEXu_u_DEFu_u_THM]: !d c b a R2 R1. happ (happ (h4/pair/LEX R1 R2) (h4/pair/_2C a b)) (h4/pair/_2C c d) <=> happ (happ R1 a) c \/ a = c /\ happ (happ R2 b) d
% Assm [h4s_setu_u_relations_finiteu_u_prefixesu_u_def]: !_0. (!e r e_27. happ (happ (happ _0 e) r) e_27 = h4/pair/_2C e_27 (h4/bool/IN (h4/pair/_2C e_27 e) r)) ==> (!s r. h4/set__relation/finite__prefixes r s <=> (!e. h4/bool/IN e s ==> h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 e) r))))
% Assm [h4s_predu_u_sets_GSPECu_u_AND]: !_1. (!P x. happ (happ _1 P) x = h4/pair/_2C x (happ P x)) ==> (!_0. (!P Q x. ?v. (v <=> happ P x /\ happ Q x) /\ happ (happ (happ _0 P) Q) x = h4/pair/_2C x v) ==> (!Q P. h4/pred__set/GSPEC (happ (happ _0 P) Q) = h4/pred__set/INTER (h4/pred__set/GSPEC (happ _1 P)) (h4/pred__set/GSPEC (happ _1 Q))))
% Assm [h4s_predu_u_sets_GSPECu_u_ID]: !_0. (!y x. happ (happ _0 y) x = h4/pair/_2C x (h4/bool/IN x y)) ==> (!y. h4/pred__set/GSPEC (happ _0 y) = y)
% Assm [h4s_predu_u_sets_INTERu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/FINITE (h4/pred__set/INTER s t))
% 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_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_combins_Su_u_DEF]: !x x x. h4/combin/S x x x = happ (happ x x) (happ x x)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_primu_u_recs_WFu_u_IFFu_u_WELLFOUNDED]: !R. h4/relation/WF R <=> h4/prim__rec/wellfounded R
% Assm [h4s_primu_u_recs_wellfoundedu_u_def]: !R. h4/prim__rec/wellfounded R <=> ~(?f. !n. happ (happ R (happ f (h4/num/SUC n))) (happ f n))
% Assm [h4s_relations_invu_u_DEF]: !y x R. happ (happ (h4/relation/inv R) x) y <=> happ (happ R y) x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_predu_u_sets_INu_u_BIGUNION]: !x sos. h4/bool/IN x (h4/pred__set/BIGUNION sos) <=> (?s. h4/bool/IN x s /\ h4/bool/IN s sos)
% Assm [h4s_predu_u_sets_FINITEu_u_BIGUNIONu_u_EQ]: !P. h4/pred__set/FINITE (h4/pred__set/BIGUNION P) <=> h4/pred__set/FINITE P /\ (!s. h4/bool/IN s P ==> h4/pred__set/FINITE s)
% Assm [h4s_predu_u_sets_IMAGEu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm [h4s_predu_u_sets_FINITEu_u_INSERT]: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% 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_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_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_relations_RTCu_u_CASES1]: !y x R. happ (happ (h4/relation/RTC R) x) y <=> x = y \/ (?u. happ (happ R x) u /\ happ (happ (h4/relation/RTC R) u) y)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% 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)
% Goal: !tc_27 r. (!x y. h4/bool/IN (h4/pair/_2C x y) r ==> happ (happ tc_27 x) y) /\ (!x y. (?z. h4/bool/IN (h4/pair/_2C x z) (h4/set__relation/tc r) /\ happ (happ tc_27 x) z /\ h4/bool/IN (h4/pair/_2C z y) r) ==> happ (happ tc_27 x) y) ==> (!x y. h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r) ==> happ (happ tc_27 x) y)
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_Q1223578,TV_Q1223574]: ![V_f, V_g]: (![V_x]: s(TV_Q1223574,happ(s(t_fun(TV_Q1223578,TV_Q1223574),V_f),s(TV_Q1223578,V_x))) = s(TV_Q1223574,happ(s(t_fun(TV_Q1223578,TV_Q1223574),V_g),s(TV_Q1223578,V_x))) => s(t_fun(TV_Q1223578,TV_Q1223574),V_f) = s(t_fun(TV_Q1223578,TV_Q1223574),V_g))).
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_TRUTH, axiom, 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_setu_u_relations_tcu_u_def, axiom, ![TV_u_27a]: ![V_x, V_x0]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_x))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_x0)))) <=> ![V_tcu_27]: (![V_a00]: ((?[V_x1, V_y]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a00) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_y))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_x),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_y))))))) | ?[V_x1, V_y]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a00) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_y))) & ?[V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_z)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_x0))))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_setu_u_relations_tcu_u_ind, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y]: (?[V_z]: (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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))) & 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_tcu_27),s(TV_u_27a,V_z))),s(TV_u_27a,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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => 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_tcu_27),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_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_setu_u_relations_tcu_u_rulesu_c0, axiom, ![TV_u_27a]: ![V_y, V_x, V_r]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) => p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))))).
fof(ah4s_setu_u_relations_tcu_u_indu_u_right, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y]: (?[V_z]: (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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_setu_u_relations_tcu_u_strongind, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y]: (?[V_z]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) & (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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))) & (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) & 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_tcu_27),s(TV_u_27a,V_z))),s(TV_u_27a,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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_setu_u_relations_tcu_u_rulesu_c1, axiom, ![TV_u_27a]: ![V_y, V_x, V_r]: (?[V_z]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))) => p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))))).
fof(ah4s_setu_u_relations_tcu_u_indu_u_left, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y]: (?[V_z]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & 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_tcu_27),s(TV_u_27a,V_z))),s(TV_u_27a,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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_setu_u_relations_rtcu_u_indu_u_right, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_domain(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_range(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_x))))) & ![V_x, V_y]: (?[V_z]: (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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_setu_u_relations_tcu_u_strongindu_u_left, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y]: (?[V_z]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) & 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_tcu_27),s(TV_u_27a,V_z))),s(TV_u_27a,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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_setu_u_relations_tcu_u_casesu_u_right, axiom, ![TV_u_27a]: ![V_y, V_x, V_r]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) <=> (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) | ?[V_z]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))))).
fof(ah4s_setu_u_relations_tcu_u_cases, axiom, ![TV_u_27a]: ![V_y, V_x, V_r]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) <=> (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) | ?[V_z]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))))))).
fof(ah4s_setu_u_relations_tcu_u_casesu_u_left, axiom, ![TV_u_27a]: ![V_y, V_x, V_r]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) <=> (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) | ?[V_z]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))))))).
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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_setu_u_relations_tcu_u_strongind0, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))))) & ![V_x, V_y]: (?[V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z)))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z)))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))))) => ![V_a0]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a0))))))).
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_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_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_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_setu_u_relations_rrestrictu_u_def, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_r, V_x, V_s, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & (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(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_bool,V_v)))) => ![V_uu_0]: (![V_r, V_s, V_x]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),V_s))) => ![V_s, V_r]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_fun(TV_u_27a,t_bool),V_s)))))))))).
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_pairs_LAMBDAu_u_PROD, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_p1, V_p2]: 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)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))),s(TV_u_27b,V_p2))) = s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_p1),s(TV_u_27b,V_p2))))) => ![V_uu_0]: (![V_P, V_p1]: s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))) = s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))) => ![V_P, V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) = s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)))))).
fof(ah4s_pairs_PFORALLu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x, V_y]: 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_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) => ![V_uu_0]: (![V_P, 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_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,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_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))) => ![V_P]: (![V_x, V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> p(s(t_bool,d_forall(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P)))))))))))).
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_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_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_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,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_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_setu_u_relations_strictu_u_def, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_r, V_x, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))) & s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_bool,V_v)))) => ![V_uu_0]: (![V_r, V_x]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(TV_u_27a,V_x))) => ![V_r]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_strict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))))))).
fof(ah4s_setu_u_relations_inu_u_rrestrict, axiom, ![TV_u_27a]: ![V_y, V_x, V_s, V_r]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & (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)))))))).
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_setu_u_relations_tcu_u_rules0u_c1, axiom, ![TV_u_27a]: ![V_y, V_x, V_r]: (?[V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))))).
fof(ah4s_setu_u_relations_tcu_u_cases0, axiom, ![TV_u_27a]: ![V_r, V_a0]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a0)))) <=> (?[V_x, V_y]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a0) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))))) | ?[V_x, V_y]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a0) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))) & ?[V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))))))))))).
fof(ah4s_setu_u_relations_tcu_u_rules0u_c0, axiom, ![TV_u_27a]: ![V_y, V_x, V_r]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))))).
fof(ah4s_setu_u_relations_tcu_u_ind0, axiom, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))))) & ![V_x, V_y]: (?[V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))))) => ![V_a0]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_tcu_27),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_a0))))))).
fof(ah4s_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t)).
fof(ah4s_bools_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_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_setu_u_relations_univu_u_relnu_u_def, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_x1, V_xs, V_x2]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x1),s(t_fun(TV_u_27a,t_bool),V_xs)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x2),s(t_fun(TV_u_27a,t_bool),V_xs)))))) & s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_1),s(TV_u_27a,V_x1))),s(t_fun(TV_u_27a,t_bool),V_xs))),s(TV_u_27a,V_x2))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27a,V_x2))),s(t_bool,V_v)))) => ![V_uu_0]: (![V_xs, V_x1]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_xs))),s(TV_u_27a,V_x1))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_1),s(TV_u_27a,V_x1))),s(t_fun(TV_u_27a,t_bool),V_xs))) => ![V_xs]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_univu_u_reln(s(t_fun(TV_u_27a,t_bool),V_xs))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_xs)))))))))).
fof(ah4s_setu_u_relations_domainu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_r, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_y]: p(s(t_bool,h4s_bools_in(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_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_r]: s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_domain(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))))).
fof(ah4s_setu_u_relations_rangeu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_r, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_x]: p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(t_bool,V_v)))) => ![V_r]: s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_range(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))))))).
fof(ah4s_setu_u_relations_finiteu_u_prefixesu_u_subset, axiom, ![TV_u_27a]: ![V_su_27, V_s, V_r]: ((p(s(t_bool,h4s_setu_u_relations_finiteu_u_prefixes(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_su_27),s(t_fun(TV_u_27a,t_bool),V_s))))) => (p(s(t_bool,h4s_setu_u_relations_finiteu_u_prefixes(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_fun(TV_u_27a,t_bool),V_su_27)))) & p(s(t_bool,h4s_setu_u_relations_finiteu_u_prefixes(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_fun(TV_u_27a,t_bool),V_su_27))),s(t_fun(TV_u_27a,t_bool),V_su_27))))))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_setu_u_relations_rrestrictu_u_rrestrict, axiom, ![TV_u_27a]: ![V_y, V_x, V_r]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_fun(TV_u_27a,t_bool),V_x))),s(t_fun(TV_u_27a,t_bool),V_y))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(TV_u_27a,t_bool),V_y)))))).
fof(ah4s_setu_u_relations_rrestrictu_u_union, axiom, ![TV_u_27a]: ![V_s, V_r2, V_r1]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r2))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r1),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r2),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_setu_u_relations_strictu_u_rrestrict, axiom, ![TV_u_27a]: ![V_s, V_r]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_strict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_rrestrict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_strict(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_fun(TV_u_27a,t_bool),V_s)))).
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_pairs_LEXu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_3]: (![V_R1, V_s, V_u, V_R2, V_t, V_v]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(TV_u_27a,V_u))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))),s(TV_u_27b,V_v)))) <=> (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_R1),s(TV_u_27a,V_s))),s(TV_u_27a,V_u)))) | (s(TV_u_27a,V_s) = s(TV_u_27a,V_u) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_t))),s(TV_u_27b,V_v))))))) => ![V_uu_2]: (![V_R1, V_s, V_R2, V_t, V_u]: 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(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))),s(TV_u_27a,V_u))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(TV_u_27a,V_u))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))) => ![V_uu_1]: (![V_R1, V_s, V_R2, V_t]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))))) => ![V_uu_0]: (![V_R1, V_R2, V_s]: s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27a,V_s))) = s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))) => ![V_R2, V_R1]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),h4s_pairs_lex(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2)))))))))).
fof(ah4s_predu_u_sets_partitionu_u_def, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_s, V_R, V_x, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) & 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_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(t_bool,V_v)))) => ![V_uu_0]: (![V_s, V_R, V_t]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & s(t_fun(TV_u_27a,t_bool),V_t) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))))))) & s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_t),s(t_bool,V_v)))) => ![V_s, V_R]: s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_partition(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))).
fof(ah4s_predu_u_sets_KoenigsLemma, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_R, V_x, V_y]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),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_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => ![V_R]: (![V_x]: 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(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x)))))))) => ![V_x]: (~ (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(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))),s(TV_u_27a,V_x))))))))) => ?[V_f]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_x) & ![V_n]: 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_R),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))))))))).
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_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_predu_u_sets_KoenigsLemmau_u_WF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_R, V_x, V_y]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),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_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => ![V_R]: ((![V_x]: 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(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x)))))))) & p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))) => ![V_x]: 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(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))),s(TV_u_27a,V_x))))))))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_LEXu_u_DEFu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_d, V_c, V_b, V_a, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),h4s_pairs_lex(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),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_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_c),s(TV_u_27b,V_d)))))) <=> (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_R1),s(TV_u_27a,V_a))),s(TV_u_27a,V_c)))) | (s(TV_u_27a,V_a) = s(TV_u_27a,V_c) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_b))),s(TV_u_27b,V_d)))))))).
fof(ah4s_setu_u_relations_finiteu_u_prefixesu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_e, V_r, V_eu_27]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27b,V_e))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))),s(TV_u_27a,V_eu_27))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_eu_27),s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_eu_27),s(TV_u_27b,V_e))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))) => ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_finiteu_u_prefixes(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r),s(t_fun(TV_u_27b,t_bool),V_s)))) <=> ![V_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_e),s(t_fun(TV_u_27b,t_bool),V_s)))) => 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_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27b,V_e))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r)))))))))))).
fof(ah4s_predu_u_sets_GSPECu_u_AND, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_P, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_P, V_Q, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_Q))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_Q, V_P]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_Q))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_Q)))))))))).
fof(ah4s_predu_u_sets_GSPECu_u_ID, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_y, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_y))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_y))))) => ![V_y]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_y))))) = s(t_fun(TV_u_27a,t_bool),V_y))).
fof(ah4s_predu_u_sets_INTERu_u_FINITE, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_t]: p(s(t_bool,h4s_predu_u_sets_finite(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)))))))).
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_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
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),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_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1))) = 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_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
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_primu_u_recs_WFu_u_IFFu_u_WELLFOUNDED, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_bool,h4s_primu_u_recs_wellfounded(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_primu_u_recs_wellfoundedu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_primu_u_recs_wellfounded(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ~ (?[V_f]: ![V_n]: 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_R),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))))))).
fof(ah4s_relations_invu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: 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)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = 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_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_predu_u_sets_INu_u_BIGUNION, axiom, ![TV_u_27a]: ![V_x, V_sos]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sos)))))) <=> ?[V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sos))))))).
fof(ah4s_predu_u_sets_FINITEu_u_BIGUNIONu_u_EQ, axiom, ![TV_u_27a]: ![V_P]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))) <=> (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_IMAGEu_u_FINITE, axiom, ![TV_u_27b,TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_FINITEu_u_INSERT, axiom, ![TV_u_27a]: ![V_x, V_s]: s(t_bool,h4s_predu_u_sets_finite(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_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))).
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_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_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_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_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) & ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_RTCu_u_CASES1, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),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_u]: (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_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_u)))) & 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)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_u))),s(TV_u_27a,V_y)))))))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ch4s_setu_u_relations_tcu_u_strongindu_u_right, conjecture, ![TV_u_27a]: ![V_tcu_27, V_r]: ((![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y]: (?[V_z]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) & (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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_x, V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_setu_u_relations_tc(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))) => 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_tcu_27),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
