%   ORIGINAL: h4/set__relation/finite__linear__order__has__maximal
% 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/set__relation/finite__acyclic__has__maximal: !s. h4/pred__set/FINITE s ==> ~(s = h4/pred__set/EMPTY) ==> (!r. h4/set__relation/acyclic r ==> (?x. h4/bool/IN x (h4/set__relation/maximal__elements s r)))
% Assm: h4/set__relation/finite__strict__linear__order__has__maximal: !s r. h4/pred__set/FINITE s /\ h4/set__relation/strict__linear__order r s /\ ~(s = h4/pred__set/EMPTY) ==> (?x. h4/bool/IN x (h4/set__relation/maximal__elements s r))
% Assm: h4/set__relation/finite__acyclic__has__maximal__path: !x s r. h4/pred__set/FINITE s /\ h4/set__relation/acyclic r /\ h4/bool/IN x s /\ ~h4/bool/IN x (h4/set__relation/maximal__elements s r) ==> (?y. h4/bool/IN y (h4/set__relation/maximal__elements s r) /\ h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r))
% Assm: h4/set__relation/maximal__union: !s r2 r1 e. h4/bool/IN e (h4/set__relation/maximal__elements s (h4/pred__set/UNION r1 r2)) ==> h4/bool/IN e (h4/set__relation/maximal__elements s r1) /\ h4/bool/IN e (h4/set__relation/maximal__elements s r2)
% Assm: h4/set__relation/maximal__elements__def: !xs r. h4/set__relation/maximal__elements xs r = h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN x xs /\ (!x_27. h4/bool/IN x_27 xs /\ h4/bool/IN (h4/pair/_2C x x_27) r ==> x = x_27)))
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/set__relation/linear__order__def: !s r. h4/set__relation/linear__order r s <=> h4/pred__set/SUBSET (h4/set__relation/domain r) s /\ h4/pred__set/SUBSET (h4/set__relation/range r) s /\ h4/set__relation/transitive r /\ h4/set__relation/antisym r /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> h4/bool/IN (h4/pair/_2C x y) r \/ h4/bool/IN (h4/pair/_2C y x) r)
% 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/set__relation/maximal__TC: !s r. h4/pred__set/SUBSET (h4/set__relation/domain r) s /\ h4/pred__set/SUBSET (h4/set__relation/range r) s ==> h4/set__relation/maximal__elements s (h4/set__relation/tc r) = h4/set__relation/maximal__elements s r
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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__CLAUSES_c0: !t. T /\ t <=> t
% 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/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/set__relation/extend__linear__order: !x s r. ~h4/bool/IN x s /\ h4/set__relation/linear__order r s ==> h4/set__relation/linear__order (h4/pred__set/UNION r (h4/pred__set/GSPEC (\y. h4/pair/_2C (h4/pair/_2C y x) (h4/bool/IN y (h4/pred__set/UNION s (h4/pred__set/INSERT x h4/pred__set/EMPTY)))))) (h4/pred__set/UNION s (h4/pred__set/INSERT x h4/pred__set/EMPTY))
% Assm: h4/set__relation/linear__order__dom__rng: !y x s r. h4/bool/IN (h4/pair/_2C x y) r /\ h4/set__relation/linear__order r s ==> h4/bool/IN x s /\ h4/bool/IN y s
% Assm: h4/set__relation/partial__order__linear__order: !s r. h4/set__relation/linear__order r s ==> h4/set__relation/partial__order r s
% Assm: h4/set__relation/strict__linear__order0: !s r. h4/set__relation/linear__order r s ==> h4/set__relation/strict__linear__order (h4/set__relation/strict r) s
% Assm: h4/set__relation/linear__order__subset: !s_27 s r. h4/set__relation/linear__order r s /\ h4/pred__set/SUBSET s_27 s ==> h4/set__relation/linear__order (h4/set__relation/rrestrict r s_27) s_27
% Assm: h4/set__relation/linear__order__restrict: !s_27 s r. h4/set__relation/linear__order r s ==> h4/set__relation/linear__order (h4/set__relation/rrestrict r s_27) (h4/pred__set/INTER s s_27)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> 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/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/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/set__relation/empty__linear__order: !r. h4/set__relation/linear__order r h4/pred__set/EMPTY <=> r = h4/pred__set/EMPTY
% Assm: h4/set__relation/linear__order0: !s r. h4/set__relation/strict__linear__order r s ==> h4/set__relation/linear__order (h4/pred__set/UNION r (h4/pred__set/GSPEC (\x. h4/pair/_2C (h4/pair/_2C x x) (h4/bool/IN x s)))) s
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/set__relation/transitive__def: !r. h4/set__relation/transitive r <=> (!x y z. h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN (h4/pair/_2C y z) r ==> h4/bool/IN (h4/pair/_2C x z) r)
% Assm: h4/set__relation/antisym__def: !r. h4/set__relation/antisym r <=> (!x y. h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN (h4/pair/_2C y x) r ==> x = y)
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/set__relation/strict__linear__order__acyclic: !s r. h4/set__relation/strict__linear__order r s ==> h4/set__relation/acyclic r
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q 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__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/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% 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/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/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> 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/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/set__relation/acyclic__def: !r. h4/set__relation/acyclic r <=> (!x. ~h4/bool/IN (h4/pair/_2C x x) (h4/set__relation/tc r))
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% 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/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/pred__set/NOT__INSERT__EMPTY: !x s. ~(h4/pred__set/INSERT x s = h4/pred__set/EMPTY)
% Assm: h4/set__relation/reflexive__def: !s r. h4/set__relation/reflexive r s <=> (!x. h4/bool/IN x s ==> h4/bool/IN (h4/pair/_2C x x) r)
% Assm: h4/set__relation/partial__order__def: !s r. h4/set__relation/partial__order r s <=> h4/pred__set/SUBSET (h4/set__relation/domain r) s /\ h4/pred__set/SUBSET (h4/set__relation/range r) s /\ h4/set__relation/transitive r /\ h4/set__relation/reflexive r s /\ h4/set__relation/antisym r
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/NOT__IMP: !B A. ~(A ==> B) <=> A /\ ~B
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% 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/set__relation/strict__linear__order__def: !s r. h4/set__relation/strict__linear__order r s <=> h4/pred__set/SUBSET (h4/set__relation/domain r) s /\ h4/pred__set/SUBSET (h4/set__relation/range r) s /\ h4/set__relation/transitive r /\ (!x. ~h4/bool/IN (h4/pair/_2C x x) r) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s /\ ~(x = y) ==> h4/bool/IN (h4/pair/_2C x y) r \/ h4/bool/IN (h4/pair/_2C y x) r)
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/pred__set/SUBSET__INTER: !u t s. h4/pred__set/SUBSET s (h4/pred__set/INTER t u) <=> h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET s u
% 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/set__relation/finite__acyclic__has__minimal: !s. h4/pred__set/FINITE s ==> ~(s = h4/pred__set/EMPTY) ==> (!r. h4/set__relation/acyclic r ==> (?x. h4/bool/IN x (h4/set__relation/minimal__elements s r)))
% 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/finite__strict__linear__order__has__minimal: !s r. h4/pred__set/FINITE s /\ h4/set__relation/strict__linear__order r s /\ ~(s = h4/pred__set/EMPTY) ==> (?x. h4/bool/IN x (h4/set__relation/minimal__elements s r))
% Assm: h4/set__relation/finite__acyclic__has__minimal__path: !x s r. h4/pred__set/FINITE s /\ h4/set__relation/acyclic r /\ h4/bool/IN x s /\ ~h4/bool/IN x (h4/set__relation/minimal__elements s r) ==> (?y. h4/bool/IN y (h4/set__relation/minimal__elements s r) /\ h4/bool/IN (h4/pair/_2C y x) (h4/set__relation/tc r))
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/pred__set/SUBSET__EMPTY: !s. h4/pred__set/SUBSET s h4/pred__set/EMPTY <=> s = h4/pred__set/EMPTY
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% 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/FINITE__CROSS__EQ: !Q P. h4/pred__set/FINITE (h4/pred__set/CROSS P Q) <=> P = h4/pred__set/EMPTY \/ Q = h4/pred__set/EMPTY \/ h4/pred__set/FINITE P /\ h4/pred__set/FINITE Q
% Assm: h4/set__relation/minimal__elements__def: !xs r. h4/set__relation/minimal__elements xs r = h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN x xs /\ (!x_27. h4/bool/IN x_27 xs /\ h4/bool/IN (h4/pair/_2C x_27 x) r ==> x = x_27)))
% 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/CROSS__SINGS: !y x. h4/pred__set/CROSS (h4/pred__set/INSERT x h4/pred__set/EMPTY) (h4/pred__set/INSERT y h4/pred__set/EMPTY) = h4/pred__set/INSERT (h4/pair/_2C x y) h4/pred__set/EMPTY
% Assm: h4/pred__set/FINITE__CROSS: !Q P. h4/pred__set/FINITE P /\ h4/pred__set/FINITE Q ==> h4/pred__set/FINITE (h4/pred__set/CROSS P Q)
% Assm: h4/pred__set/CROSS__SUBSET: !Q0 Q P0 P. h4/pred__set/SUBSET (h4/pred__set/CROSS P0 Q0) (h4/pred__set/CROSS P Q) <=> P0 = h4/pred__set/EMPTY \/ Q0 = h4/pred__set/EMPTY \/ h4/pred__set/SUBSET P0 P /\ h4/pred__set/SUBSET Q0 Q
% Assm: h4/pred__set/CROSS__INSERT__RIGHT: !x Q P. h4/pred__set/CROSS P (h4/pred__set/INSERT x Q) = h4/pred__set/UNION (h4/pred__set/CROSS P (h4/pred__set/INSERT x h4/pred__set/EMPTY)) (h4/pred__set/CROSS P Q)
% Assm: h4/pred__set/CROSS__INSERT__LEFT: !x Q P. h4/pred__set/CROSS (h4/pred__set/INSERT x P) Q = h4/pred__set/UNION (h4/pred__set/CROSS (h4/pred__set/INSERT x h4/pred__set/EMPTY) Q) (h4/pred__set/CROSS P Q)
% Assm: h4/pred__set/IN__CROSS: !x Q P. h4/bool/IN x (h4/pred__set/CROSS P Q) <=> h4/bool/IN (h4/pair/FST x) P /\ h4/bool/IN (h4/pair/SND x) Q
% Assm: h4/pred__set/CROSS__EMPTY_c0: !P. h4/pred__set/CROSS P h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm: h4/pred__set/FINITE__COMPLETE__INDUCTION: !P. (!x. (!y. h4/pred__set/PSUBSET y x ==> P y) ==> h4/pred__set/FINITE x ==> P x) ==> (!x. h4/pred__set/FINITE x ==> P x)
% Goal: !s r. h4/pred__set/FINITE s /\ h4/set__relation/linear__order r s /\ ~(s = h4/pred__set/EMPTY) ==> (?x. h4/bool/IN x (h4/set__relation/maximal__elements s r))
%   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_setu_u_relations_finiteu_u_acyclicu_u_hasu_u_maximal]: !s. h4/pred__set/FINITE s ==> ~(s = h4/pred__set/EMPTY) ==> (!r. h4/set__relation/acyclic r ==> (?x. h4/bool/IN x (h4/set__relation/maximal__elements s r)))
% Assm [h4s_setu_u_relations_finiteu_u_strictu_u_linearu_u_orderu_u_hasu_u_maximal]: !s r. h4/pred__set/FINITE s /\ h4/set__relation/strict__linear__order r s /\ ~(s = h4/pred__set/EMPTY) ==> (?x. h4/bool/IN x (h4/set__relation/maximal__elements s r))
% Assm [h4s_setu_u_relations_finiteu_u_acyclicu_u_hasu_u_maximalu_u_path]: !x s r. h4/pred__set/FINITE s /\ h4/set__relation/acyclic r /\ h4/bool/IN x s /\ ~h4/bool/IN x (h4/set__relation/maximal__elements s r) ==> (?y. h4/bool/IN y (h4/set__relation/maximal__elements s r) /\ h4/bool/IN (h4/pair/_2C x y) (h4/set__relation/tc r))
% Assm [h4s_setu_u_relations_maximalu_u_union]: !s r2 r1 e. h4/bool/IN e (h4/set__relation/maximal__elements s (h4/pred__set/UNION r1 r2)) ==> h4/bool/IN e (h4/set__relation/maximal__elements s r1) /\ h4/bool/IN e (h4/set__relation/maximal__elements s r2)
% Assm [h4s_setu_u_relations_maximalu_u_elementsu_u_def]: !_0. (!xs r x. ?v. (v <=> h4/bool/IN x xs /\ (!x_27. h4/bool/IN x_27 xs /\ h4/bool/IN (h4/pair/_2C x x_27) r ==> x = x_27)) /\ happ (happ (happ _0 xs) r) x = h4/pair/_2C x v) ==> (!xs r. h4/set__relation/maximal__elements xs r = h4/pred__set/GSPEC (happ (happ _0 xs) r))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_setu_u_relations_linearu_u_orderu_u_def]: !s r. h4/set__relation/linear__order r s <=> h4/pred__set/SUBSET (h4/set__relation/domain r) s /\ h4/pred__set/SUBSET (h4/set__relation/range r) s /\ h4/set__relation/transitive r /\ h4/set__relation/antisym r /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> h4/bool/IN (h4/pair/_2C x y) r \/ h4/bool/IN (h4/pair/_2C y x) r)
% 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_setu_u_relations_maximalu_u_TC]: !s r. h4/pred__set/SUBSET (h4/set__relation/domain r) s /\ h4/pred__set/SUBSET (h4/set__relation/range r) s ==> h4/set__relation/maximal__elements s (h4/set__relation/tc r) = h4/set__relation/maximal__elements s r
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_setu_u_relations_extendu_u_linearu_u_order]: !_0. (!s x y. happ (happ (happ _0 s) x) y = h4/pair/_2C (h4/pair/_2C y x) (h4/bool/IN y (h4/pred__set/UNION s (h4/pred__set/INSERT x h4/pred__set/EMPTY)))) ==> (!x s r. ~h4/bool/IN x s /\ h4/set__relation/linear__order r s ==> h4/set__relation/linear__order (h4/pred__set/UNION r (h4/pred__set/GSPEC (happ (happ _0 s) x))) (h4/pred__set/UNION s (h4/pred__set/INSERT x h4/pred__set/EMPTY)))
% Assm [h4s_setu_u_relations_linearu_u_orderu_u_domu_u_rng]: !y x s r. h4/bool/IN (h4/pair/_2C x y) r /\ h4/set__relation/linear__order r s ==> h4/bool/IN x s /\ h4/bool/IN y s
% Assm [h4s_setu_u_relations_partialu_u_orderu_u_linearu_u_order]: !s r. h4/set__relation/linear__order r s ==> h4/set__relation/partial__order r s
% Assm [h4s_setu_u_relations_strictu_u_linearu_u_order0]: !s r. h4/set__relation/linear__order r s ==> h4/set__relation/strict__linear__order (h4/set__relation/strict r) s
% Assm [h4s_setu_u_relations_linearu_u_orderu_u_subset]: !s_27 s r. h4/set__relation/linear__order r s /\ h4/pred__set/SUBSET s_27 s ==> h4/set__relation/linear__order (h4/set__relation/rrestrict r s_27) s_27
% Assm [h4s_setu_u_relations_linearu_u_orderu_u_restrict]: !s_27 s r. h4/set__relation/linear__order r s ==> h4/set__relation/linear__order (h4/set__relation/rrestrict r s_27) (h4/pred__set/INTER s s_27)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~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_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> 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_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_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_setu_u_relations_emptyu_u_linearu_u_order]: !r. h4/set__relation/linear__order r h4/pred__set/EMPTY <=> r = h4/pred__set/EMPTY
% Assm [h4s_setu_u_relations_linearu_u_order0]: !_0. (!s x. happ (happ _0 s) x = h4/pair/_2C (h4/pair/_2C x x) (h4/bool/IN x s)) ==> (!s r. h4/set__relation/strict__linear__order r s ==> h4/set__relation/linear__order (h4/pred__set/UNION r (h4/pred__set/GSPEC (happ _0 s))) s)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% 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_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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_setu_u_relations_transitiveu_u_def]: !r. h4/set__relation/transitive r <=> (!x y z. h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN (h4/pair/_2C y z) r ==> h4/bool/IN (h4/pair/_2C x z) r)
% Assm [h4s_setu_u_relations_antisymu_u_def]: !r. h4/set__relation/antisym r <=> (!x y. h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN (h4/pair/_2C y x) r ==> x = y)
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_setu_u_relations_strictu_u_linearu_u_orderu_u_acyclic]: !s r. h4/set__relation/strict__linear__order r s ==> h4/set__relation/acyclic r
% 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_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_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_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% 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_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_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> 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_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_setu_u_relations_acyclicu_u_def]: !r. h4/set__relation/acyclic r <=> (!x. ~h4/bool/IN (h4/pair/_2C x x) (h4/set__relation/tc r))
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% 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_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% 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_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_predu_u_sets_NOTu_u_INSERTu_u_EMPTY]: !x s. ~(h4/pred__set/INSERT x s = h4/pred__set/EMPTY)
% Assm [h4s_setu_u_relations_reflexiveu_u_def]: !s r. h4/set__relation/reflexive r s <=> (!x. h4/bool/IN x s ==> h4/bool/IN (h4/pair/_2C x x) r)
% Assm [h4s_setu_u_relations_partialu_u_orderu_u_def]: !s r. h4/set__relation/partial__order r s <=> h4/pred__set/SUBSET (h4/set__relation/domain r) s /\ h4/pred__set/SUBSET (h4/set__relation/range r) s /\ h4/set__relation/transitive r /\ h4/set__relation/reflexive r s /\ h4/set__relation/antisym r
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% 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_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% 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_setu_u_relations_strictu_u_linearu_u_orderu_u_def]: !s r. h4/set__relation/strict__linear__order r s <=> h4/pred__set/SUBSET (h4/set__relation/domain r) s /\ h4/pred__set/SUBSET (h4/set__relation/range r) s /\ h4/set__relation/transitive r /\ (!x. ~h4/bool/IN (h4/pair/_2C x x) r) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s /\ ~(x = y) ==> h4/bool/IN (h4/pair/_2C x y) r \/ h4/bool/IN (h4/pair/_2C y x) r)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_predu_u_sets_SUBSETu_u_INTER]: !u t s. h4/pred__set/SUBSET s (h4/pred__set/INTER t u) <=> h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET s u
% 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_setu_u_relations_finiteu_u_acyclicu_u_hasu_u_minimal]: !s. h4/pred__set/FINITE s ==> ~(s = h4/pred__set/EMPTY) ==> (!r. h4/set__relation/acyclic r ==> (?x. h4/bool/IN x (h4/set__relation/minimal__elements s r)))
% 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_finiteu_u_strictu_u_linearu_u_orderu_u_hasu_u_minimal]: !s r. h4/pred__set/FINITE s /\ h4/set__relation/strict__linear__order r s /\ ~(s = h4/pred__set/EMPTY) ==> (?x. h4/bool/IN x (h4/set__relation/minimal__elements s r))
% Assm [h4s_setu_u_relations_finiteu_u_acyclicu_u_hasu_u_minimalu_u_path]: !x s r. h4/pred__set/FINITE s /\ h4/set__relation/acyclic r /\ h4/bool/IN x s /\ ~h4/bool/IN x (h4/set__relation/minimal__elements s r) ==> (?y. h4/bool/IN y (h4/set__relation/minimal__elements s r) /\ h4/bool/IN (h4/pair/_2C y x) (h4/set__relation/tc r))
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_predu_u_sets_SUBSETu_u_EMPTY]: !s. h4/pred__set/SUBSET s h4/pred__set/EMPTY <=> s = h4/pred__set/EMPTY
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% 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_FINITEu_u_CROSSu_u_EQ]: !Q P. h4/pred__set/FINITE (h4/pred__set/CROSS P Q) <=> P = h4/pred__set/EMPTY \/ Q = h4/pred__set/EMPTY \/ h4/pred__set/FINITE P /\ h4/pred__set/FINITE Q
% Assm [h4s_setu_u_relations_minimalu_u_elementsu_u_def]: !_0. (!xs r x. ?v. (v <=> h4/bool/IN x xs /\ (!x_27. h4/bool/IN x_27 xs /\ h4/bool/IN (h4/pair/_2C x_27 x) r ==> x = x_27)) /\ happ (happ (happ _0 xs) r) x = h4/pair/_2C x v) ==> (!xs r. h4/set__relation/minimal__elements xs r = h4/pred__set/GSPEC (happ (happ _0 xs) r))
% 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_CROSSu_u_SINGS]: !y x. h4/pred__set/CROSS (h4/pred__set/INSERT x h4/pred__set/EMPTY) (h4/pred__set/INSERT y h4/pred__set/EMPTY) = h4/pred__set/INSERT (h4/pair/_2C x y) h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_FINITEu_u_CROSS]: !Q P. h4/pred__set/FINITE P /\ h4/pred__set/FINITE Q ==> h4/pred__set/FINITE (h4/pred__set/CROSS P Q)
% Assm [h4s_predu_u_sets_CROSSu_u_SUBSET]: !Q0 Q P0 P. h4/pred__set/SUBSET (h4/pred__set/CROSS P0 Q0) (h4/pred__set/CROSS P Q) <=> P0 = h4/pred__set/EMPTY \/ Q0 = h4/pred__set/EMPTY \/ h4/pred__set/SUBSET P0 P /\ h4/pred__set/SUBSET Q0 Q
% Assm [h4s_predu_u_sets_CROSSu_u_INSERTu_u_RIGHT]: !x Q P. h4/pred__set/CROSS P (h4/pred__set/INSERT x Q) = h4/pred__set/UNION (h4/pred__set/CROSS P (h4/pred__set/INSERT x h4/pred__set/EMPTY)) (h4/pred__set/CROSS P Q)
% Assm [h4s_predu_u_sets_CROSSu_u_INSERTu_u_LEFT]: !x Q P. h4/pred__set/CROSS (h4/pred__set/INSERT x P) Q = h4/pred__set/UNION (h4/pred__set/CROSS (h4/pred__set/INSERT x h4/pred__set/EMPTY) Q) (h4/pred__set/CROSS P Q)
% Assm [h4s_predu_u_sets_INu_u_CROSS]: !x Q P. h4/bool/IN x (h4/pred__set/CROSS P Q) <=> h4/bool/IN (h4/pair/FST x) P /\ h4/bool/IN (h4/pair/SND x) Q
% Assm [h4s_predu_u_sets_CROSSu_u_EMPTYu_c0]: !P. h4/pred__set/CROSS P h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_FINITEu_u_COMPLETEu_u_INDUCTION]: !P. (!x. (!y. h4/pred__set/PSUBSET y x ==> happ P y) ==> h4/pred__set/FINITE x ==> happ P x) ==> (!x. h4/pred__set/FINITE x ==> happ P x)
% Goal: !s r. h4/pred__set/FINITE s /\ h4/set__relation/linear__order r s /\ ~(s = h4/pred__set/EMPTY) ==> (?x. h4/bool/IN x (h4/set__relation/maximal__elements s r))
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_Q1224940,TV_Q1224936]: ![V_f, V_g]: (![V_x]: s(TV_Q1224936,happ(s(t_fun(TV_Q1224940,TV_Q1224936),V_f),s(TV_Q1224940,V_x))) = s(TV_Q1224936,happ(s(t_fun(TV_Q1224940,TV_Q1224936),V_g),s(TV_Q1224940,V_x))) => s(t_fun(TV_Q1224940,TV_Q1224936),V_f) = s(t_fun(TV_Q1224940,TV_Q1224936),V_g))).
fof(ah4s_setu_u_relations_finiteu_u_acyclicu_u_hasu_u_maximal, 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)))) => (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => ![V_r]: (p(s(t_bool,h4s_setu_u_relations_acyclic(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),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_maximalu_u_elements(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),V_r)))))))))).
fof(ah4s_setu_u_relations_finiteu_u_strictu_u_linearu_u_orderu_u_hasu_u_maximal, axiom, ![TV_u_27a]: ![V_s, V_r]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_setu_u_relations_strictu_u_linearu_u_order(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(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) => ?[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_maximalu_u_elements(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),V_r)))))))).
fof(ah4s_setu_u_relations_finiteu_u_acyclicu_u_hasu_u_maximalu_u_path, axiom, ![TV_u_27a]: ![V_x, V_s, V_r]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_setu_u_relations_acyclic(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_x),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_maximalu_u_elements(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),V_r)))))))))) => ?[V_y]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_maximalu_u_elements(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),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_maximalu_u_union, axiom, ![TV_u_27a]: ![V_s, V_r2, V_r1, V_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_maximalu_u_elements(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),V_r1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r2)))))))) => (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_maximalu_u_elements(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),V_r1)))))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_maximalu_u_elements(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),V_r2))))))))).
fof(ah4s_setu_u_relations_maximalu_u_elementsu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_xs, V_r, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_xs)))) & ![V_xu_27]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_xu_27),s(t_fun(TV_u_27a,t_bool),V_xs)))) & 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_xu_27))),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_xu_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_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(t_h4s_pairs_prod(TV_u_27a,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_xs))),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_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_xs, V_r]: s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_maximalu_u_elements(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),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_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(t_h4s_pairs_prod(TV_u_27a,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_xs))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))).
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_linearu_u_orderu_u_def, axiom, ![TV_u_27a]: ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_linearu_u_order(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),h4s_setu_u_relations_domain(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),h4s_setu_u_relations_range(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_setu_u_relations_transitive(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & (p(s(t_bool,h4s_setu_u_relations_antisym(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s))))) => (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_y),s(TV_u_27a,V_x))),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_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_setu_u_relations_maximalu_u_TC, axiom, ![TV_u_27a]: ![V_s, V_r]: ((p(s(t_bool,h4s_predu_u_sets_subset(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))),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),h4s_setu_u_relations_range(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(TV_u_27a,t_bool),h4s_setu_u_relations_maximalu_u_elements(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_tc(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_setu_u_relations_maximalu_u_elements(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),V_r))))).
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_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_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_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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, 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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_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_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_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_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_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_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_setu_u_relations_extendu_u_linearu_u_order, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_s, V_x, 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(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_s))),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_y),s(TV_u_27a,V_x))),s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),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_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))))))) => ![V_x, V_s, 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_setu_u_relations_linearu_u_order(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_setu_u_relations_linearu_u_order(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_r),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_gspec(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_s))),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),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_setu_u_relations_linearu_u_orderu_u_domu_u_rng, 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),V_r)))) & p(s(t_bool,h4s_setu_u_relations_linearu_u_order(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(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_setu_u_relations_partialu_u_orderu_u_linearu_u_order, axiom, ![TV_u_27a]: ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_linearu_u_order(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_setu_u_relations_partialu_u_order(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_setu_u_relations_strictu_u_linearu_u_order0, axiom, ![TV_u_27a]: ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_linearu_u_order(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_setu_u_relations_strictu_u_linearu_u_order(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_setu_u_relations_linearu_u_orderu_u_subset, axiom, ![TV_u_27a]: ![V_su_27, V_s, V_r]: ((p(s(t_bool,h4s_setu_u_relations_linearu_u_order(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_linearu_u_order(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_setu_u_relations_linearu_u_orderu_u_restrict, axiom, ![TV_u_27a]: ![V_su_27, V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_linearu_u_order(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_setu_u_relations_linearu_u_order(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),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_su_27)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_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_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_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_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_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_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_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_setu_u_relations_emptyu_u_linearu_u_order, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_setu_u_relations_linearu_u_order(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_empty)))) <=> 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_empty))).
fof(ah4s_setu_u_relations_linearu_u_order0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_s, V_x]: 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))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))) = 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_x))),s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_strictu_u_linearu_u_order(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_setu_u_relations_linearu_u_order(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_r),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_gspec(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))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_s))))))),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_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_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_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_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_setu_u_relations_transitiveu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_setu_u_relations_transitive(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) <=> ![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_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_y),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_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))).
fof(ah4s_setu_u_relations_antisymu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_setu_u_relations_antisym(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),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,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(TV_u_27a,V_x))),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)))).
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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_setu_u_relations_strictu_u_linearu_u_orderu_u_acyclic, axiom, ![TV_u_27a]: ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_strictu_u_linearu_u_order(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_setu_u_relations_acyclic(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))).
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_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_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_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_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_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_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_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_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_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_setu_u_relations_acyclicu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_setu_u_relations_acyclic(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) <=> ![V_x]: ~ (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_x))),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_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_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_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_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
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_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_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_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_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_setu_u_relations_reflexiveu_u_def, axiom, ![TV_u_27a]: ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_reflexive(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)))) <=> ![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(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_x))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))).
fof(ah4s_setu_u_relations_partialu_u_orderu_u_def, axiom, ![TV_u_27a]: ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_partialu_u_order(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),h4s_setu_u_relations_domain(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),h4s_setu_u_relations_range(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_setu_u_relations_transitive(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & (p(s(t_bool,h4s_setu_u_relations_reflexive(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_setu_u_relations_antisym(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))))))).
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_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_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_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_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_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_setu_u_relations_strictu_u_linearu_u_orderu_u_def, axiom, ![TV_u_27a]: ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_strictu_u_linearu_u_order(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),h4s_setu_u_relations_domain(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),h4s_setu_u_relations_range(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_setu_u_relations_transitive(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & (![V_x]: ~ (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_x))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) & ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,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,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(TV_u_27a,V_x))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_SUBSETu_u_INTER, axiom, ![TV_u_27a]: ![V_u, 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),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u)))))) <=> (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_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_u))))))).
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_setu_u_relations_finiteu_u_acyclicu_u_hasu_u_minimal, 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)))) => (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => ![V_r]: (p(s(t_bool,h4s_setu_u_relations_acyclic(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),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_minimalu_u_elements(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),V_r)))))))))).
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_finiteu_u_strictu_u_linearu_u_orderu_u_hasu_u_minimal, axiom, ![TV_u_27a]: ![V_s, V_r]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_setu_u_relations_strictu_u_linearu_u_order(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(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) => ?[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_minimalu_u_elements(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),V_r)))))))).
fof(ah4s_setu_u_relations_finiteu_u_acyclicu_u_hasu_u_minimalu_u_path, axiom, ![TV_u_27a]: ![V_x, V_s, V_r]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_setu_u_relations_acyclic(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_x),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_minimalu_u_elements(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),V_r)))))))))) => ?[V_y]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_minimalu_u_elements(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),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_y),s(TV_u_27a,V_x))),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_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(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_predu_u_sets_SUBSETu_u_EMPTY, 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_empty)))) <=> 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_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_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_FINITEu_u_CROSSu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_Q, V_P]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> (s(t_fun(TV_u_27a,t_bool),V_P) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty) | (s(t_fun(TV_u_27b,t_bool),V_Q) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty) | (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_Q))))))))).
fof(ah4s_setu_u_relations_minimalu_u_elementsu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_xs, V_r, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_xs)))) & ![V_xu_27]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_xu_27),s(t_fun(TV_u_27a,t_bool),V_xs)))) & 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_xu_27),s(TV_u_27a,V_x))),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_xu_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_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(t_h4s_pairs_prod(TV_u_27a,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_xs))),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_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_xs, V_r]: s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_minimalu_u_elements(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),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_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(t_h4s_pairs_prod(TV_u_27a,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_xs))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))).
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_CROSSu_u_SINGS, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(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_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty))))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_insert(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),h4s_predu_u_sets_empty)))).
fof(ah4s_predu_u_sets_FINITEu_u_CROSS, axiom, ![TV_u_27a,TV_u_27b]: ![V_Q, V_P]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_Q))))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))))).
fof(ah4s_predu_u_sets_CROSSu_u_SUBSET, axiom, ![TV_u_27a,TV_u_27b]: ![V_Q0, V_Q, V_P0, V_P]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P0),s(t_fun(TV_u_27b,t_bool),V_Q0))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> (s(t_fun(TV_u_27a,t_bool),V_P0) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty) | (s(t_fun(TV_u_27b,t_bool),V_Q0) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty) | (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_P0),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27b,t_bool),V_Q0),s(t_fun(TV_u_27b,t_bool),V_Q))))))))).
fof(ah4s_predu_u_sets_CROSSu_u_INSERTu_u_RIGHT, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_Q, V_P]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_bool),V_Q))))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))).
fof(ah4s_predu_u_sets_CROSSu_u_INSERTu_u_LEFT, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_Q, V_P]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(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_P))),s(t_fun(TV_u_27b,t_bool),V_Q))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(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_27b,t_bool),V_Q))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))).
fof(ah4s_predu_u_sets_INu_u_CROSS, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_Q, V_P]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
fof(ah4s_predu_u_sets_CROSSu_u_EMPTYu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_FINITEu_u_COMPLETEu_u_INDUCTION, axiom, ![TV_u_27a]: ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_y),s(t_fun(TV_u_27a,t_bool),V_x)))) => 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_y))))) => (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_x)))) => 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_x)))))) => ![V_x]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_x)))) => 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_x))))))).
fof(ch4s_setu_u_relations_finiteu_u_linearu_u_orderu_u_hasu_u_maximal, conjecture, ![TV_u_27a]: ![V_s, V_r]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_setu_u_relations_linearu_u_order(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(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) => ?[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_maximalu_u_elements(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),V_r)))))))).
