%   ORIGINAL: h4/pred__set/infinite__rest
% 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/pred__set/FINITE__REST: !s. h4/pred__set/FINITE s ==> h4/pred__set/FINITE (h4/pred__set/REST s)
% Assm: h4/pred__set/REST__DEF: !s. h4/pred__set/REST s = h4/pred__set/DELETE s (h4/pred__set/CHOICE s)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/pred__set/FINITE__DELETE: !x s. h4/pred__set/FINITE (h4/pred__set/DELETE s x) <=> h4/pred__set/FINITE s
% Assm: h4/pred__set/IN__DELETE: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm: h4/pred__set/ITSET__tupled__primitive__def: !f. h4/pred__set/ITSET__tupled f = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!b s. h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> R (h4/pair/_2C (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b)) (h4/pair/_2C s b)))) (\ITSET__tupled a. h4/pair/pair__CASE a (\s b. h4/combin/I (h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND (s = h4/pred__set/EMPTY) b (ITSET__tupled (h4/pair/_2C (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b)))) h4/bool/ARB)))
% Assm: h4/pred__set/REST__PSUBSET: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/PSUBSET (h4/pred__set/REST s) s
% Assm: h4/pred__set/CHOICE__INSERT__REST: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/INSERT (h4/pred__set/CHOICE s) (h4/pred__set/REST s) = s
% Assm: h4/pred__set/REST__SUBSET: !s. h4/pred__set/SUBSET (h4/pred__set/REST s) s
% Assm: h4/pred__set/ITSET__ind: !f P. (!s b. (h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> P (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b)) ==> P s b) ==> (!v v1. P v v1)
% Assm: h4/pred__set/ITSET__IND: !f P. (!s b. (h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> P (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b)) ==> P s b) ==> (!v v1. P v v1)
% Assm: h4/pred__set/ITSET__THM: !s f b. h4/pred__set/FINITE s ==> h4/pred__set/ITSET f s b = h4/bool/COND (s = h4/pred__set/EMPTY) b (h4/pred__set/ITSET f (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b))
% Assm: h4/pred__set/ITSET__def: !s f b. h4/pred__set/ITSET f s b = h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND (s = h4/pred__set/EMPTY) b (h4/pred__set/ITSET f (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b))) h4/bool/ARB
% Assm: h4/pred__set/CHOICE__NOT__IN__REST: !s. ~h4/bool/IN (h4/pred__set/CHOICE s) (h4/pred__set/REST s)
% Assm: h4/pred__set/SING__IFF__EMPTY__REST: !s. h4/pred__set/SING s <=> ~(s = h4/pred__set/EMPTY) /\ h4/pred__set/REST s = h4/pred__set/EMPTY
% 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/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/pred__set/REST__SING: !x. h4/pred__set/REST (h4/pred__set/INSERT x h4/pred__set/EMPTY) = h4/pred__set/EMPTY
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pred__set/ITSET__INSERT: !s. h4/pred__set/FINITE s ==> (!f x b. h4/pred__set/ITSET f (h4/pred__set/INSERT x s) b = h4/pred__set/ITSET f (h4/pred__set/REST (h4/pred__set/INSERT x s)) (f (h4/pred__set/CHOICE (h4/pred__set/INSERT x s)) b))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/pred__set/CHOICE__DEF: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/CHOICE s) s
% Assm: h4/pred__set/CHOICE__SING: !x. h4/pred__set/CHOICE (h4/pred__set/INSERT x h4/pred__set/EMPTY) = x
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% 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/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% 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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pred__set/CARD__PSUBSET: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/PSUBSET t s ==> h4/prim__rec/_3C (h4/pred__set/CARD t) (h4/pred__set/CARD s))
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/prim__rec/measure__def: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/pred__set/ITSET__curried__def: !x1 x f. h4/pred__set/ITSET f x x1 = h4/pred__set/ITSET__tupled f (h4/pair/_2C x x1)
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/pred__set/PSUBSET__DEF: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ ~(s = t)
% Assm: h4/pred__set/SING__DELETE: !x. h4/pred__set/DELETE (h4/pred__set/INSERT x h4/pred__set/EMPTY) x = h4/pred__set/EMPTY
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/NOT__IMP: !B A. ~(A ==> B) <=> A /\ ~B
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/pred__set/IN__SING: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% Assm: h4/pred__set/SING__DEF: !s. h4/pred__set/SING s <=> (?x. s = h4/pred__set/INSERT x h4/pred__set/EMPTY)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/pred__set/INFINITE__SUBSET: !s. ~h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET s t ==> ~h4/pred__set/FINITE t)
% Assm: h4/pred__set/FINITE__PSUBSET__INFINITE: !s. ~h4/pred__set/FINITE s <=> (!t. h4/pred__set/FINITE t ==> h4/pred__set/SUBSET t s ==> h4/pred__set/PSUBSET t s)
% Assm: h4/pred__set/NOT__IN__FINITE: ~h4/pred__set/FINITE h4/pred__set/UNIV <=> (!s. h4/pred__set/FINITE s ==> (?x. ~h4/bool/IN x s))
% Assm: h4/pred__set/SUBSET__FINITE: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t)
% Assm: h4/pred__set/IN__INFINITE__NOT__FINITE: !t s. ~h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> (?x. h4/bool/IN x s /\ ~h4/bool/IN x t)
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/pred__set/FINITE__PSUBSET__UNIV: ~h4/pred__set/FINITE h4/pred__set/UNIV <=> (!s. h4/pred__set/FINITE s ==> h4/pred__set/PSUBSET s h4/pred__set/UNIV)
% Assm: h4/pred__set/CARD__BIGUNION__SAME__SIZED__SETS: !s n. h4/pred__set/FINITE s /\ (!e. h4/bool/IN e s ==> h4/pred__set/FINITE e /\ h4/pred__set/CARD e = n) /\ (!e1 e2. h4/bool/IN e1 s /\ h4/bool/IN e2 s /\ ~(e1 = e2) ==> h4/pred__set/DISJOINT e1 e2) ==> h4/pred__set/CARD (h4/pred__set/BIGUNION s) = h4/arithmetic/_2A (h4/pred__set/CARD s) n
% Assm: h4/pred__set/SUBSET__REFL: !s. h4/pred__set/SUBSET s s
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/pred__set/NOT__INSERT__EMPTY: !x s. ~(h4/pred__set/INSERT x s = h4/pred__set/EMPTY)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/pred__set/EQ__UNIV: !s. (!x. h4/bool/IN x s) <=> s = h4/pred__set/UNIV
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/SUBSET__UNIV: !s. h4/pred__set/SUBSET s h4/pred__set/UNIV
% Assm: h4/pred__set/FINITE__EMPTY: h4/pred__set/FINITE h4/pred__set/EMPTY
% Assm: h4/pred__set/FINITE__DEF: !s. h4/pred__set/FINITE s <=> (!P. P h4/pred__set/EMPTY /\ (!s0. P s0 ==> (!e. P (h4/pred__set/INSERT e s0))) ==> P s)
% Assm: h4/pred__set/ABSORPTION: !x s. h4/bool/IN x s <=> h4/pred__set/INSERT x s = s
% Assm: h4/pred__set/IN__BIGUNION: !x sos. h4/bool/IN x (h4/pred__set/BIGUNION sos) <=> (?s. h4/bool/IN x s /\ h4/bool/IN s sos)
% Assm: h4/pred__set/BIGUNION__EMPTY: h4/pred__set/BIGUNION h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/pred__set/BIGUNION__INSERT: !s P. h4/pred__set/BIGUNION (h4/pred__set/INSERT s P) = h4/pred__set/UNION s (h4/pred__set/BIGUNION P)
% Assm: h4/pred__set/CARD__UNION__EQN: !t s. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/CARD (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/CARD s) (h4/pred__set/CARD t)) (h4/pred__set/CARD (h4/pred__set/INTER s t))
% Assm: h4/pred__set/CARD__INSERT: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
% Assm: h4/pred__set/CARD__EMPTY: h4/pred__set/CARD h4/pred__set/EMPTY = h4/num/0
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% 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/DISJOINT__DEF: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/INTER s t = h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/arithmetic/MULT__0: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/SUB__0_c1: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm: h4/arithmetic/MULT__COMM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/arithmetic/MULT__CLAUSES_c4: !n m. h4/arithmetic/_2A (h4/num/SUC m) n = h4/arithmetic/_2B (h4/arithmetic/_2A m n) n
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/bool/RIGHT__FORALL__IMP__THM: !Q P. (!x. P ==> Q x) <=> P ==> (!x. Q x)
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% 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/OR__CLAUSES_c0: !t. T \/ t <=> T
% Goal: !s. ~h4/pred__set/FINITE s ==> ~h4/pred__set/FINITE (h4/pred__set/REST s)
%   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_predu_u_sets_FINITEu_u_REST]: !s. h4/pred__set/FINITE s ==> h4/pred__set/FINITE (h4/pred__set/REST s)
% Assm [h4s_predu_u_sets_RESTu_u_DEF]: !s. h4/pred__set/REST s = h4/pred__set/DELETE s (h4/pred__set/CHOICE s)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_predu_u_sets_FINITEu_u_DELETE]: !x s. h4/pred__set/FINITE (h4/pred__set/DELETE s x) <=> h4/pred__set/FINITE s
% Assm [h4s_predu_u_sets_INu_u_DELETE]: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm [h4s_predu_u_sets_ITSETu_u_tupledu_u_primitiveu_u_def]: !_4. (!ITSET__tupled f s b. ?v. (v <=> s = h4/pred__set/EMPTY) /\ happ (happ (happ (happ _4 ITSET__tupled) f) s) b = h4/combin/I (h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND v b (happ ITSET__tupled (h4/pair/_2C (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b)))) h4/bool/ARB)) ==> (!_3. (!ITSET__tupled f s. happ (happ (happ _3 ITSET__tupled) f) s = happ (happ (happ _4 ITSET__tupled) f) s) ==> (!_2. (!ITSET__tupled f a. happ (happ (happ _2 ITSET__tupled) f) a = h4/pair/pair__CASE a (happ (happ _3 ITSET__tupled) f)) ==> (!_1. (!f ITSET__tupled. happ (happ _1 f) ITSET__tupled = happ (happ _2 ITSET__tupled) f) ==> (!_0. (!f R. happ (happ _0 f) R <=> h4/relation/WF R /\ (!b s. h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> happ (happ R (h4/pair/_2C (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b))) (h4/pair/_2C s b))) ==> (!f. h4/pred__set/ITSET__tupled f = h4/relation/WFREC (h4/min/_40 (happ _0 f)) (happ _1 f))))))
% Assm [h4s_predu_u_sets_RESTu_u_PSUBSET]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/PSUBSET (h4/pred__set/REST s) s
% Assm [h4s_predu_u_sets_CHOICEu_u_INSERTu_u_REST]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/INSERT (h4/pred__set/CHOICE s) (h4/pred__set/REST s) = s
% Assm [h4s_predu_u_sets_RESTu_u_SUBSET]: !s. h4/pred__set/SUBSET (h4/pred__set/REST s) s
% Assm [h4s_predu_u_sets_ITSETu_u_ind]: !f P. (!s b. (h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> happ (happ P (h4/pred__set/REST s)) (happ (happ f (h4/pred__set/CHOICE s)) b)) ==> happ (happ P s) b) ==> (!v v1. happ (happ P v) v1)
% Assm [h4s_predu_u_sets_ITSETu_u_IND]: !f P. (!s b. (h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> happ (happ P (h4/pred__set/REST s)) (happ (happ f (h4/pred__set/CHOICE s)) b)) ==> happ (happ P s) b) ==> (!v v1. happ (happ P v) v1)
% Assm [h4s_predu_u_sets_ITSETu_u_THM]: !s f b. h4/pred__set/FINITE s ==> (?v. (v <=> s = h4/pred__set/EMPTY) /\ h4/pred__set/ITSET f s b = h4/bool/COND v b (h4/pred__set/ITSET f (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b)))
% Assm [h4s_predu_u_sets_ITSETu_u_def]: !s f b. ?v. (v <=> s = h4/pred__set/EMPTY) /\ h4/pred__set/ITSET f s b = h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND v b (h4/pred__set/ITSET f (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b))) h4/bool/ARB
% Assm [h4s_predu_u_sets_CHOICEu_u_NOTu_u_INu_u_REST]: !s. ~h4/bool/IN (h4/pred__set/CHOICE s) (h4/pred__set/REST s)
% Assm [h4s_predu_u_sets_SINGu_u_IFFu_u_EMPTYu_u_REST]: !s. h4/pred__set/SING s <=> ~(s = h4/pred__set/EMPTY) /\ h4/pred__set/REST s = h4/pred__set/EMPTY
% 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_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_predu_u_sets_RESTu_u_SING]: !x. h4/pred__set/REST (h4/pred__set/INSERT x h4/pred__set/EMPTY) = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_ITSETu_u_INSERT]: !s. h4/pred__set/FINITE s ==> (!f x b. h4/pred__set/ITSET f (h4/pred__set/INSERT x s) b = h4/pred__set/ITSET f (h4/pred__set/REST (h4/pred__set/INSERT x s)) (happ (happ f (h4/pred__set/CHOICE (h4/pred__set/INSERT x s))) b))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_predu_u_sets_CHOICEu_u_DEF]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/CHOICE s) s
% Assm [h4s_predu_u_sets_CHOICEu_u_SING]: !x. h4/pred__set/CHOICE (h4/pred__set/INSERT x h4/pred__set/EMPTY) = x
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% 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_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% 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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_predu_u_sets_CARDu_u_PSUBSET]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/PSUBSET t s ==> happ (happ h4/prim__rec/_3C (h4/pred__set/CARD t)) (h4/pred__set/CARD s))
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (happ h4/prim__rec/measure m)
% Assm [h4s_primu_u_recs_measureu_u_def]: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (happ (h4/relation/inv__image R) f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_combins_ou_u_DEF]: !g f x. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_predu_u_sets_ITSETu_u_curriedu_u_def]: !x1 x f. h4/pred__set/ITSET f x x1 = happ (h4/pred__set/ITSET__tupled f) (h4/pair/_2C x x1)
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_predu_u_sets_PSUBSETu_u_DEF]: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ ~(s = t)
% Assm [h4s_predu_u_sets_SINGu_u_DELETE]: !x. h4/pred__set/DELETE (h4/pred__set/INSERT x h4/pred__set/EMPTY) x = h4/pred__set/EMPTY
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_predu_u_sets_INu_u_SING]: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% Assm [h4s_predu_u_sets_SINGu_u_DEF]: !s. h4/pred__set/SING s <=> (?x. s = h4/pred__set/INSERT x h4/pred__set/EMPTY)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_predu_u_sets_INFINITEu_u_SUBSET]: !s. ~h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET s t ==> ~h4/pred__set/FINITE t)
% Assm [h4s_predu_u_sets_FINITEu_u_PSUBSETu_u_INFINITE]: !s. ~h4/pred__set/FINITE s <=> (!t. h4/pred__set/FINITE t ==> h4/pred__set/SUBSET t s ==> h4/pred__set/PSUBSET t s)
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_FINITE]: ~h4/pred__set/FINITE h4/pred__set/UNIV <=> (!s. h4/pred__set/FINITE s ==> (?x. ~h4/bool/IN x s))
% Assm [h4s_predu_u_sets_SUBSETu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t)
% Assm [h4s_predu_u_sets_INu_u_INFINITEu_u_NOTu_u_FINITE]: !t s. ~h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> (?x. h4/bool/IN x s /\ ~h4/bool/IN x t)
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_predu_u_sets_FINITEu_u_PSUBSETu_u_UNIV]: ~h4/pred__set/FINITE h4/pred__set/UNIV <=> (!s. h4/pred__set/FINITE s ==> h4/pred__set/PSUBSET s h4/pred__set/UNIV)
% Assm [h4s_predu_u_sets_CARDu_u_BIGUNIONu_u_SAMEu_u_SIZEDu_u_SETS]: !s n. h4/pred__set/FINITE s /\ (!e. h4/bool/IN e s ==> h4/pred__set/FINITE e /\ h4/pred__set/CARD e = n) /\ (!e1 e2. h4/bool/IN e1 s /\ h4/bool/IN e2 s /\ ~(e1 = e2) ==> h4/pred__set/DISJOINT e1 e2) ==> h4/pred__set/CARD (h4/pred__set/BIGUNION s) = h4/arithmetic/_2A (h4/pred__set/CARD s) n
% Assm [h4s_predu_u_sets_SUBSETu_u_REFL]: !s. h4/pred__set/SUBSET s s
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_predu_u_sets_NOTu_u_INSERTu_u_EMPTY]: !x s. ~(h4/pred__set/INSERT x s = h4/pred__set/EMPTY)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_predu_u_sets_EQu_u_UNIV]: !s. (!x. h4/bool/IN x s) <=> s = h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_SUBSETu_u_UNIV]: !s. h4/pred__set/SUBSET s h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_FINITEu_u_EMPTY]: h4/pred__set/FINITE h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_FINITEu_u_DEF]: !s. h4/pred__set/FINITE s <=> (!P. happ P h4/pred__set/EMPTY /\ (!s0. happ P s0 ==> (!e. happ P (h4/pred__set/INSERT e s0))) ==> happ P s)
% Assm [h4s_predu_u_sets_ABSORPTION]: !x s. h4/bool/IN x s <=> h4/pred__set/INSERT x s = s
% Assm [h4s_predu_u_sets_INu_u_BIGUNION]: !x sos. h4/bool/IN x (h4/pred__set/BIGUNION sos) <=> (?s. h4/bool/IN x s /\ h4/bool/IN s sos)
% Assm [h4s_predu_u_sets_BIGUNIONu_u_EMPTY]: h4/pred__set/BIGUNION h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_predu_u_sets_BIGUNIONu_u_INSERT]: !s P. h4/pred__set/BIGUNION (h4/pred__set/INSERT s P) = h4/pred__set/UNION s (h4/pred__set/BIGUNION P)
% Assm [h4s_predu_u_sets_CARDu_u_UNIONu_u_EQN]: !t s. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/CARD (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/CARD s) (h4/pred__set/CARD t)) (h4/pred__set/CARD (h4/pred__set/INTER s t))
% Assm [h4s_predu_u_sets_CARDu_u_INSERT]: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
% Assm [h4s_predu_u_sets_CARDu_u_EMPTY]: h4/pred__set/CARD h4/pred__set/EMPTY = h4/num/0
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% 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_DISJOINTu_u_DEF]: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/INTER s t = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_arithmetics_MULTu_u_0]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_0u_c1]: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm [h4s_arithmetics_MULTu_u_COMM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c4]: !n m. h4/arithmetic/_2A (h4/num/SUC m) n = h4/arithmetic/_2B (h4/arithmetic/_2A m n) n
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. P ==> happ Q x) <=> P ==> (!x. happ Q x)
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_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_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Goal: !s. ~h4/pred__set/FINITE s ==> ~h4/pred__set/FINITE (h4/pred__set/REST s)
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_Q1243994,TV_Q1243990]: ![V_f, V_g]: (![V_x]: s(TV_Q1243990,happ(s(t_fun(TV_Q1243994,TV_Q1243990),V_f),s(TV_Q1243994,V_x))) = s(TV_Q1243990,happ(s(t_fun(TV_Q1243994,TV_Q1243990),V_g),s(TV_Q1243994,V_x))) => s(t_fun(TV_Q1243994,TV_Q1243990),V_f) = s(t_fun(TV_Q1243994,TV_Q1243990),V_g))).
fof(ah4s_predu_u_sets_FINITEu_u_REST, 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)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_RESTu_u_DEF, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_predu_u_sets_FINITEu_u_DELETE, 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_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))))) = 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_DELETE, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_predu_u_sets_ITSETu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_4]: (![V_ITSETu_u_tupled, V_f, V_s, V_b]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,V_b))) = s(TV_u_27b,h4s_combins_i(s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))))),s(TV_u_27b,h4s_bools_arb)))))) => ![V_uu_3]: (![V_ITSETu_u_tupled, V_f, V_s]: s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))) => ![V_uu_2]: (![V_ITSETu_u_tupled, V_f, V_a]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),V_a))) = s(TV_u_27b,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),V_a),s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))) => ![V_uu_1]: (![V_f, V_ITSETu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))) => ![V_uu_0]: (![V_f, V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R)))) & ![V_b, V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27b,V_b))))))))) => ![V_f]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),h4s_predu_u_sets_itsetu_u_tupled(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))))))))).
fof(ah4s_predu_u_sets_RESTu_u_PSUBSET, axiom, ![TV_u_27a]: ![V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_CHOICEu_u_INSERTu_u_REST, axiom, ![TV_u_27a]: ![V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_predu_u_sets_RESTu_u_SUBSET, axiom, ![TV_u_27a]: ![V_s]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_predu_u_sets_ITSETu_u_ind, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_P]: (![V_s, V_b]: (((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,V_b))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_bool),V_v))),s(TV_u_27b,V_v1)))))).
fof(ah4s_predu_u_sets_ITSETu_u_IND, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_P]: (![V_s, V_b]: (((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,V_b))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_bool),V_v))),s(TV_u_27b,V_v1)))))).
fof(ah4s_predu_u_sets_ITSETu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_f, V_b]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27b,V_b))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b)))))))))).
fof(ah4s_predu_u_sets_ITSETu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_f, V_b]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27b,V_b))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))),s(TV_u_27b,h4s_bools_arb))))).
fof(ah4s_predu_u_sets_CHOICEu_u_NOTu_u_INu_u_REST, axiom, ![TV_u_27a]: ![V_s]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_SINGu_u_IFFu_u_EMPTYu_u_REST, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_sing(s(t_fun(TV_u_27a,t_bool),V_s)))) <=> (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))).
fof(ah4s_predu_u_sets_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_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> 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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_RESTu_u_SING, axiom, ![TV_u_27a]: ![V_x]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_predu_u_sets_ITSETu_u_INSERT, axiom, ![TV_u_27a,TV_u_27b]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f, V_x, V_b]: s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,V_b))) = s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(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(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(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(TV_u_27b,V_b))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_CHOICEu_u_DEF, axiom, ![TV_u_27a]: ![V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_CHOICEu_u_SING, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_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_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_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_bools_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_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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_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_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(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,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_predu_u_sets_CARDu_u_PSUBSET, 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_psubset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_t))))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),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_y))))) => 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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(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_27a,V_x)).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_primu_u_recs_WFu_u_measure, axiom, ![TV_u_27a]: ![V_m]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure),s(t_fun(TV_u_27a,t_h4s_nums_num),V_m))))))).
fof(ah4s_primu_u_recs_measureu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure) = s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c)))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_x)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: 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)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = 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_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, 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_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
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,h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_predu_u_sets_ITSETu_u_curriedu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x1, V_x, V_f]: s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27b,V_x1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),h4s_predu_u_sets_itsetu_u_tupled(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27b,V_x1)))))).
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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> (p(s(t_bool,h4s_predu_u_sets_subset(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),V_s) = s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_predu_u_sets_SINGu_u_DELETE, axiom, ![TV_u_27a]: ![V_x]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_IMP, 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_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_INu_u_SING, axiom, ![TV_u_27a]: ![V_y, V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_predu_u_sets_SINGu_u_DEF, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_sing(s(t_fun(TV_u_27a,t_bool),V_s)))) <=> ?[V_x]: s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_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_predu_u_sets_INFINITEu_u_SUBSET, 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_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) => ~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_predu_u_sets_FINITEu_u_PSUBSETu_u_INFINITE, 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),V_t)))) => (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_FINITE, axiom, ![TV_u_27a]: (~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))) <=> ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ?[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_SUBSETu_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_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_INu_u_INFINITEu_u_NOTu_u_FINITE, axiom, ![TV_u_27a]: ![V_t, V_s]: ((~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s))))) & p(s(t_bool,h4s_predu_u_sets_finite(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_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_FINITEu_u_PSUBSETu_u_UNIV, axiom, ![TV_u_27a]: (~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))) <=> ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))))).
fof(ah4s_predu_u_sets_CARDu_u_BIGUNIONu_u_SAMEu_u_SIZEDu_u_SETS, axiom, ![TV_u_27a]: ![V_s, V_n]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) & (![V_e]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_e),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) => (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_e)))) & s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_e))) = s(t_h4s_nums_num,V_n))) & ![V_e1, V_e2]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_e1),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_e2),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) & ~ (s(t_fun(TV_u_27a,t_bool),V_e1) = s(t_fun(TV_u_27a,t_bool),V_e2)))) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_e1),s(t_fun(TV_u_27a,t_bool),V_e2))))))) => s(t_h4s_nums_num,h4s_predu_u_sets_card(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_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s))),s(t_h4s_nums_num,V_n))))).
fof(ah4s_predu_u_sets_SUBSETu_u_REFL, axiom, ![TV_u_27a]: ![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_s))))).
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_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_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_predu_u_sets_NOTu_u_INSERTu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x, V_s]: ~ (s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
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_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_predu_u_sets_EQu_u_UNIV, axiom, ![TV_u_27a]: ![V_s]: (![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_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))).
fof(ah4s_predu_u_sets_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_predu_u_sets_SUBSETu_u_UNIV, axiom, ![TV_u_27a]: ![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),h4s_predu_u_sets_univ))))).
fof(ah4s_predu_u_sets_FINITEu_u_EMPTY, axiom, ![TV_u_27a]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))).
fof(ah4s_predu_u_sets_FINITEu_u_DEF, 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_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s0]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s0)))) => ![V_e]: p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s0)))))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_ABSORPTION, axiom, ![TV_u_27a]: ![V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_predu_u_sets_INu_u_BIGUNION, axiom, ![TV_u_27a]: ![V_x, V_sos]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sos)))))) <=> ?[V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sos))))))).
fof(ah4s_predu_u_sets_BIGUNIONu_u_EMPTY, axiom, ![TV_u_27a]: 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),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
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_predu_u_sets_BIGUNIONu_u_INSERT, axiom, ![TV_u_27a]: ![V_s, V_P]: 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),h4s_predu_u_sets_insert(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))))) = 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),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))).
fof(ah4s_predu_u_sets_CARDu_u_UNIONu_u_EQN, axiom, ![TV_u_27a]: ![V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))) => s(t_h4s_nums_num,h4s_predu_u_sets_card(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))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_t))))),s(t_h4s_nums_num,h4s_predu_u_sets_card(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_CARDu_u_INSERT, 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_x]: s(t_h4s_nums_num,h4s_predu_u_sets_card(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_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_predu_u_sets_CARDu_u_EMPTY, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_DISJOINTu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_disjoint(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_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
fof(ah4s_predu_u_sets_INu_u_INTER, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_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_arithmetics_MULTu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_SUBu_u_0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c4, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_IMPu_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_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_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_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ch4s_predu_u_sets_infiniteu_u_rest, conjecture, ![TV_u_27a]: ![V_s]: (~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s))))) => ~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
