%   ORIGINAL: h4/quantHeuristics/GUESS__RULES__ELIM__UNIT_c4
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/quantHeuristics/GUESS__RULES__ELIM__UNIT_c2: !vt i. h4/quantHeuristics/GUESS__EXISTS i vt <=> h4/quantHeuristics/GUESS__EXISTS (\x. i (h4/pair/_2C x h4/one/one0)) vt
% Assm: h4/quantHeuristics/GUESS__RULES__ELIM__UNIT_c3: !vt i. h4/quantHeuristics/GUESS__FORALL i vt <=> h4/quantHeuristics/GUESS__FORALL (\x. i (h4/pair/_2C x h4/one/one0)) vt
% Assm: h4/quantHeuristics/GUESS__RULES__ELIM__UNIT_c1: !vt i. h4/quantHeuristics/GUESS__EXISTS__POINT i vt <=> h4/quantHeuristics/GUESS__EXISTS__POINT (\x. i (h4/pair/_2C x h4/one/one0)) vt
% Assm: h4/quantHeuristics/GUESS__RULES__ELIM__UNIT_c0: !vt i. h4/quantHeuristics/GUESS__FORALL__POINT i vt <=> h4/quantHeuristics/GUESS__FORALL__POINT (\x. i (h4/pair/_2C x h4/one/one0)) vt
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/one/one__induction: !P. P h4/one/one0 ==> (!x. P x)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/quantHeuristics/GUESS__REWRITES_c4: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P <=> (!v. P v ==> (?fv. v = i fv))
% 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/quantHeuristics/GUESS__REWRITES_c3: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~P (i fv))
% Assm: h4/quantHeuristics/GUESS__REWRITES_c5: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i P <=> (!v. ~P v ==> (?fv. v = i fv))
% Assm: h4/quantHeuristics/GUESS__REWRITES_c2: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P <=> (!fv. P (i fv))
% Assm: h4/quantHeuristics/GUESS__REWRITES_c1: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~P v ==> (?fv. ~P (i fv)))
% Assm: h4/quantHeuristics/GUESS__REWRITES_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. P v ==> (?fv. P (i fv)))
% Assm: h4/pair/EXISTS__PROD: !P. (?p. P p) <=> (?p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/pair/FORALL__PROD: !P. (!p. P p) <=> (!p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/one/one1: !v. v = h4/one/one0
% Assm: h4/one/one__axiom: !g f. f = g
% Assm: h4/bool/boolAxiom: !t2 t1. ?fn. fn T = t1 /\ fn F = t2
% Assm: h4/one/one__Axiom: !e. h4/bool/_3F_21 (\fn. fn h4/one/one0 = e)
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/option/OPTION__GUARD__def_c0: h4/option/OPTION__GUARD T = h4/option/SOME h4/one/one0
% Assm: h4/one/one__case__thm: !x. h4/one/one__CASE h4/one/one0 x = x
% Assm: h4/one/one__prim__rec: !e. ?fn. fn h4/one/one0 = e
% Assm: h4/one/FORALL__ONE: !P. (!x. P x) <=> P h4/one/one0
% Assm: h4/option/OPTION__GUARD__EQ__THM_c0: !b. h4/option/OPTION__GUARD b = h4/option/SOME h4/one/one0 <=> b
% Assm: h4/one/one__DEF: h4/one/one0 = h4/min/_40 (\x. T)
% Assm: h4/option/NONE__DEF: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm: h4/option/OPTION__GUARD__COND: !b. h4/option/OPTION__GUARD b = h4/bool/COND b (h4/option/SOME h4/one/one0) h4/option/NONE
% Assm: h4/quantHeuristics/GUESS__EXISTS__GAP__def: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P <=> (!v. P v ==> (?fv. v = i fv))
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/option/OPTION__GUARD__def_c1: h4/option/OPTION__GUARD F = h4/option/NONE
% Assm: h4/one/one__case__def: !x u. h4/one/one__CASE u x = x
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/quantHeuristics/GUESSES__NEG__REWRITE_c2: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. ~P x) <=> h4/quantHeuristics/GUESS__FORALL__GAP i (\x. P x)
% Assm: h4/quantHeuristics/GUESSES__NEG__REWRITE_c3: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i (\x. ~P x) <=> h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P x)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL______NEW__FV__1_c3: !i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP (\xxx. i y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c3: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS__GAP i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c2: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__GAP i P
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/quantHeuristics/GUESS__RULES__EQUATION__EXISTS__GAP: !i. h4/quantHeuristics/GUESS__EXISTS__GAP (\xxx. i) (\x. x = i)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/quantHeuristics/GUESS__RULES__EXISTS______NEW__FV__1_c2: !i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP (\xxx. i y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. ?y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__BOOL_c2: h4/quantHeuristics/GUESS__EXISTS__GAP (\ARB. T) (\x. x)
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL______NEW__FV_c3: !iy P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP (iy y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP (\fv. iy (h4/pair/FST fv) (h4/pair/SND fv)) (\x. !y. P x y)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/quantHeuristics/GUESS__RULES__ONE__CASE______EXISTS__GAP: !Q P. (!x. ?fv. x = Q fv) ==> h4/quantHeuristics/GUESS__EXISTS__GAP Q P
% Assm: h4/quantHeuristics/GUESSES__NEG__REWRITE_c5: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i (\x. ~P x) <=> h4/quantHeuristics/GUESS__EXISTS__POINT i (\x. P x)
% Assm: h4/quantHeuristics/GUESSES__NEG__REWRITE_c1: !i P. h4/quantHeuristics/GUESS__FORALL i (\x. ~P x) <=> h4/quantHeuristics/GUESS__EXISTS i (\x. P x)
% Assm: h4/quantHeuristics/GUESSES__UEXISTS__THM2: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP (\x. i) P ==> (h4/bool/_3F_21 P <=> P i)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/quantHeuristics/GUESSES__WEAKEN__THM_c3: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P ==> h4/quantHeuristics/GUESS__EXISTS i P
% Assm: h4/quantHeuristics/GUESS__RULES__EXISTS______NEW__FV_c2: !iy P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP (iy y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP (\fv. iy (h4/pair/FST fv) (h4/pair/SND fv)) (\x. ?y. P x y)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/quantHeuristics/GUESS__RULES__EQUIV_c6: !i P2 P1. h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P1 x) /\ h4/quantHeuristics/GUESS__FORALL__GAP i (\x. P2 x) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P1 x <=> P2 x)
% Assm: h4/quantHeuristics/GUESS__RULES__EQUIV_c7: !i P2 P1. h4/quantHeuristics/GUESS__FORALL__GAP i (\x. P1 x) /\ h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P2 x) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P1 x <=> P2 x)
% Assm: h4/quantHeuristics/GUESS__RULES__EQUIV_c5: !i P2 P1. h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P1 x) /\ h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P2 x) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (\x. P1 x <=> P2 x)
% Assm: h4/quantHeuristics/GUESSES__UEXISTS__THM4: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT (\x. i) P ==> h4/quantHeuristics/GUESS__EXISTS__GAP (\x. i) P ==> (h4/bool/_3F_21 P <=> T)
% Assm: h4/quantHeuristics/GUESS__RULES__NEG_c1: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P x) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (\x. ~P x)
% Assm: h4/quantHeuristics/GUESS__RULES__EXISTS_c2: !i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. ?y. P x y)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c4: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__POINT i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c5: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS__POINT i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c1: !i P. h4/quantHeuristics/GUESS__FORALL i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS i P
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL______NEW__FV__1_c2: !i P. (!y. h4/quantHeuristics/GUESS__FORALL__GAP (\xxx. i y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL______NEW__FV__1_c1: !i P. (!y. h4/quantHeuristics/GUESS__FORALL (\xxx. i y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL i (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL______NEW__FV__1_c0: !i P. (!y. h4/quantHeuristics/GUESS__FORALL__POINT (\xxx. i y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT i (\x. !y. P x y)
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL______NEW__FV_c2: !iy P. (!y. h4/quantHeuristics/GUESS__FORALL__GAP (iy y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL__GAP (\fv. iy (h4/pair/FST fv) (h4/pair/SND fv)) (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL______NEW__FV_c1: !iy P. (!y. h4/quantHeuristics/GUESS__FORALL (iy y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL (\fv. iy (h4/pair/FST fv) (h4/pair/SND fv)) (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL______NEW__FV_c0: !iy P. (!y. h4/quantHeuristics/GUESS__FORALL__POINT (iy y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT (\fv. iy (h4/pair/FST fv) (h4/pair/SND fv)) (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESSES__NEG__REWRITE_c4: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i (\x. ~P x) <=> h4/quantHeuristics/GUESS__FORALL__POINT i (\x. P x)
% Assm: h4/quantHeuristics/GUESSES__NEG__REWRITE_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i (\x. ~P x) <=> h4/quantHeuristics/GUESS__FORALL i (\x. P x)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/LEFT__FORALL__OR__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/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/EXISTS__UNIQUE__THM: !P. h4/bool/_3F_21 (\x. P x) <=> (?x. P x) /\ (!x y. P x /\ P y ==> x = y)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL_c5: !i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL_c4: !iK P. (!y. h4/quantHeuristics/GUESS__EXISTS (\xxx. iK) (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS (\xxx. iK) (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL_c3: !i P. (!y. h4/quantHeuristics/GUESS__EXISTS__POINT i (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__POINT i (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL_c2: !i P. (!y. h4/quantHeuristics/GUESS__FORALL__GAP i (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL_c1: !i P. (!y. h4/quantHeuristics/GUESS__FORALL i (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL i (\x. !y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__FORALL_c0: !i P. (!y. h4/quantHeuristics/GUESS__FORALL__POINT i (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT i (\x. !y. P x y)
% Goal: !vt i. h4/quantHeuristics/GUESS__EXISTS__GAP i vt <=> h4/quantHeuristics/GUESS__EXISTS__GAP (\x. i (h4/pair/_2C x h4/one/one0)) vt
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c2]: !_0. (!i x. happ (happ _0 i) x = happ i (h4/pair/_2C x h4/one/one0)) ==> (!vt i. h4/quantHeuristics/GUESS__EXISTS i vt <=> h4/quantHeuristics/GUESS__EXISTS (happ _0 i) vt)
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c3]: !_0. (!i x. happ (happ _0 i) x = happ i (h4/pair/_2C x h4/one/one0)) ==> (!vt i. h4/quantHeuristics/GUESS__FORALL i vt <=> h4/quantHeuristics/GUESS__FORALL (happ _0 i) vt)
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c1]: !_0. (!i x. happ (happ _0 i) x = happ i (h4/pair/_2C x h4/one/one0)) ==> (!vt i. h4/quantHeuristics/GUESS__EXISTS__POINT i vt <=> h4/quantHeuristics/GUESS__EXISTS__POINT (happ _0 i) vt)
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c0]: !_0. (!i x. happ (happ _0 i) x = happ i (h4/pair/_2C x h4/one/one0)) ==> (!vt i. h4/quantHeuristics/GUESS__FORALL__POINT i vt <=> h4/quantHeuristics/GUESS__FORALL__POINT (happ _0 i) vt)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_ones_oneu_u_induction]: !P. happ P h4/one/one0 ==> (!x. happ P x)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c4]: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P <=> (!v. happ P v ==> (?fv. v = happ i fv))
% 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_quantHeuristicss_GUESSu_u_REWRITESu_c3]: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~happ P (happ i fv))
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c5]: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i P <=> (!v. ~happ P v ==> (?fv. v = happ i fv))
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c2]: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P <=> (!fv. happ P (happ i fv))
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c1]: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~happ P v ==> (?fv. ~happ P (happ i fv)))
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c0]: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. happ P v ==> (?fv. happ P (happ i fv)))
% Assm [h4s_pairs_EXISTSu_u_PROD]: !P. (?p. happ P p) <=> (?p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% Assm [h4s_pairs_FORALLu_u_PROD]: !P. (!p. happ P p) <=> (!p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% Assm [h4s_ones_one1]: !v. v = h4/one/one0
% Assm [h4s_ones_oneu_u_axiom]: !g f. f = g
% Assm [h4s_bools_boolAxiom]: !t2 t1. ?fn. happ fn T = t1 /\ happ fn F = t2
% Assm [h4s_ones_oneu_u_Axiom]: !_0. (!e fn. happ (happ _0 e) fn <=> happ fn h4/one/one0 = e) ==> (!e. h4/bool/_3F_21 (happ _0 e))
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_options_OPTIONu_u_GUARDu_u_defu_c0]: h4/option/OPTION__GUARD T = h4/option/SOME h4/one/one0
% Assm [h4s_ones_oneu_u_caseu_u_thm]: !x. h4/one/one__CASE h4/one/one0 x = x
% Assm [h4s_ones_oneu_u_primu_u_rec]: !e. ?fn. happ fn h4/one/one0 = e
% Assm [h4s_ones_FORALLu_u_ONE]: !P. (!x. happ P x) <=> happ P h4/one/one0
% Assm [h4s_options_OPTIONu_u_GUARDu_u_EQu_u_THMu_c0]: !b. h4/option/OPTION__GUARD b = h4/option/SOME h4/one/one0 <=> b
% Assm [h4s_ones_oneu_u_DEF]: !_0. (!x. happ _0 x <=> T) ==> h4/one/one0 = h4/min/_40 _0
% Assm [h4s_options_NONEu_u_DEF]: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm [h4s_options_OPTIONu_u_GUARDu_u_COND]: !b. h4/option/OPTION__GUARD b = h4/bool/COND b (h4/option/SOME h4/one/one0) h4/option/NONE
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_GAPu_u_def]: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P <=> (!v. happ P v ==> (?fv. v = happ i fv))
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_options_OPTIONu_u_GUARDu_u_defu_c1]: h4/option/OPTION__GUARD F = h4/option/NONE
% Assm [h4s_ones_oneu_u_caseu_u_def]: !x u. h4/one/one__CASE u x = x
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c2]: !_1. (!P x. happ (happ _1 P) x <=> happ P x) ==> (!_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!i P. h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _0 P) <=> h4/quantHeuristics/GUESS__FORALL__GAP i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c3]: !_1. (!P x. happ (happ _1 P) x <=> happ P x) ==> (!_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!i P. h4/quantHeuristics/GUESS__FORALL__GAP i (happ _0 P) <=> h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_u_1u_c3]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!P y x. happ (happ (happ _1 P) y) x <=> happ (happ P x) y) ==> (!_0. (!i y xxx. happ (happ (happ _0 i) y) xxx = happ i y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP (happ (happ _0 i) y) (happ (happ _1 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c3]: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS__GAP i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c2]: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__GAP i P
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EQUATIONu_u_EXISTSu_u_GAP]: !_1. (!i x. happ (happ _1 i) x <=> x = i) ==> (!_0. (!i xxx. happ (happ _0 i) xxx = i) ==> (!i. h4/quantHeuristics/GUESS__EXISTS__GAP (happ _0 i) (happ _1 i)))
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_u_u_u_u_u_NEWu_u_FVu_u_1u_c2]: !_2. (!P x. happ (happ _2 P) x <=> (?y. happ (happ P x) y)) ==> (!_1. (!P y x. happ (happ (happ _1 P) y) x <=> happ (happ P x) y) ==> (!_0. (!i y xxx. happ (happ (happ _0 i) y) xxx = happ i y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP (happ (happ _0 i) y) (happ (happ _1 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_BOOLu_c2]: !_1. (!x. happ _1 x <=> x) ==> (!_0. (!ARB. happ _0 ARB <=> T) ==> h4/quantHeuristics/GUESS__EXISTS__GAP _0 _1)
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_c3]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!iy fv. happ (happ _1 iy) fv = happ (happ iy (h4/pair/FST fv)) (h4/pair/SND fv)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!iy P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP (happ iy y) (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP (happ _1 iy) (happ _2 P))))
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_ONEu_u_CASEu_u_u_u_u_u_EXISTSu_u_GAP]: !Q P. (!x. ?fv. x = happ Q fv) ==> h4/quantHeuristics/GUESS__EXISTS__GAP Q P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c5]: !_1. (!P x. happ (happ _1 P) x <=> happ P x) ==> (!_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!i P. h4/quantHeuristics/GUESS__FORALL__POINT i (happ _0 P) <=> h4/quantHeuristics/GUESS__EXISTS__POINT i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c1]: !_1. (!P x. happ (happ _1 P) x <=> happ P x) ==> (!_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!i P. h4/quantHeuristics/GUESS__FORALL i (happ _0 P) <=> h4/quantHeuristics/GUESS__EXISTS i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSESu_u_UEXISTSu_u_THM2]: !_0. (!i x. happ (happ _0 i) x = i) ==> (!i P. h4/quantHeuristics/GUESS__EXISTS__GAP (happ _0 i) P ==> (h4/bool/_3F_21 P <=> happ P i))
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_quantHeuristicss_GUESSESu_u_WEAKENu_u_THMu_c3]: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P ==> h4/quantHeuristics/GUESS__EXISTS i P
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_u_u_u_u_u_NEWu_u_FVu_c2]: !_2. (!P x. happ (happ _2 P) x <=> (?y. happ (happ P x) y)) ==> (!_1. (!iy fv. happ (happ _1 iy) fv = happ (happ iy (h4/pair/FST fv)) (h4/pair/SND fv)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!iy P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP (happ iy y) (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP (happ _1 iy) (happ _2 P))))
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EQUIVu_c6]: !_1. (!P1 P2 x. happ (happ (happ _1 P1) P2) x <=> happ P1 x <=> happ P2 x) ==> (!_0. (!P1 x. happ (happ _0 P1) x <=> happ P1 x) ==> (!i P2 P1. h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _0 P1) /\ h4/quantHeuristics/GUESS__FORALL__GAP i (happ _0 P2) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _0 (happ (happ _1 P1) P2))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EQUIVu_c7]: !_1. (!P1 P2 x. happ (happ (happ _1 P1) P2) x <=> happ P1 x <=> happ P2 x) ==> (!_0. (!P1 x. happ (happ _0 P1) x <=> happ P1 x) ==> (!i P2 P1. h4/quantHeuristics/GUESS__FORALL__GAP i (happ _0 P1) /\ h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _0 P2) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _0 (happ (happ _1 P1) P2))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EQUIVu_c5]: !_1. (!P1 P2 x. happ (happ (happ _1 P1) P2) x <=> happ P1 x <=> happ P2 x) ==> (!_0. (!P1 x. happ (happ _0 P1) x <=> happ P1 x) ==> (!i P2 P1. h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _0 P1) /\ h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _0 P2) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (happ _0 (happ (happ _1 P1) P2))))
% Assm [h4s_quantHeuristicss_GUESSESu_u_UEXISTSu_u_THM4]: !_1. (!i x. happ (happ _1 i) x = i) ==> (!_0. (!i x. happ (happ _0 i) x = i) ==> (!i P. h4/quantHeuristics/GUESS__EXISTS__POINT (happ _0 i) P ==> h4/quantHeuristics/GUESS__EXISTS__GAP (happ _1 i) P ==> (h4/bool/_3F_21 P <=> T)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_NEGu_c1]: !_1. (!P x. happ (happ _1 P) x <=> ~happ P x) ==> (!_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!i P. h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _0 P) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (happ _0 (happ _1 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_c2]: !_1. (!P x. happ (happ _1 P) x <=> (?y. happ (happ P x) y)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _1 P)))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c4]: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__POINT i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c5]: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS__POINT i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c0]: !i P. h4/quantHeuristics/GUESS__EXISTS i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c1]: !i P. h4/quantHeuristics/GUESS__FORALL i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS i P
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_u_1u_c2]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!P y x. happ (happ (happ _1 P) y) x <=> happ (happ P x) y) ==> (!_0. (!i y xxx. happ (happ (happ _0 i) y) xxx = happ i y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__FORALL__GAP (happ (happ _0 i) y) (happ (happ _1 P) y)) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_u_1u_c1]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!P y x. happ (happ (happ _1 P) y) x <=> happ (happ P x) y) ==> (!_0. (!i y xxx. happ (happ (happ _0 i) y) xxx = happ i y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__FORALL (happ (happ _0 i) y) (happ (happ _1 P) y)) ==> h4/quantHeuristics/GUESS__FORALL i (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_u_1u_c0]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!P y x. happ (happ (happ _1 P) y) x <=> happ (happ P x) y) ==> (!_0. (!i y xxx. happ (happ (happ _0 i) y) xxx = happ i y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__FORALL__POINT (happ (happ _0 i) y) (happ (happ _1 P) y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT i (happ _2 P))))
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_c2]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!iy fv. happ (happ _1 iy) fv = happ (happ iy (h4/pair/FST fv)) (h4/pair/SND fv)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!iy P. (!y. h4/quantHeuristics/GUESS__FORALL__GAP (happ iy y) (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__FORALL__GAP (happ _1 iy) (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_c1]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!iy fv. happ (happ _1 iy) fv = happ (happ iy (h4/pair/FST fv)) (h4/pair/SND fv)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!iy P. (!y. h4/quantHeuristics/GUESS__FORALL (happ iy y) (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__FORALL (happ _1 iy) (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_c0]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!iy fv. happ (happ _1 iy) fv = happ (happ iy (h4/pair/FST fv)) (h4/pair/SND fv)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!iy P. (!y. h4/quantHeuristics/GUESS__FORALL__POINT (happ iy y) (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT (happ _1 iy) (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c4]: !_1. (!P x. happ (happ _1 P) x <=> happ P x) ==> (!_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!i P. h4/quantHeuristics/GUESS__EXISTS__POINT i (happ _0 P) <=> h4/quantHeuristics/GUESS__FORALL__POINT i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c0]: !_1. (!P x. happ (happ _1 P) x <=> happ P x) ==> (!_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!i P. h4/quantHeuristics/GUESS__EXISTS i (happ _0 P) <=> h4/quantHeuristics/GUESS__FORALL i (happ _1 P)))
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_THM]: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. h4/bool/_3F_21 (happ _0 P) <=> (?x. happ P x) /\ (!x y. happ P x /\ happ P y ==> x = y))
% 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_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c5]: !_1. (!P x. happ (happ _1 P) x <=> (!y. happ (happ P x) y)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c4]: !_2. (!P x. happ (happ _2 P) x <=> (!y. happ (happ P x) y)) ==> (!_1. (!P y x. happ (happ (happ _1 P) y) x <=> happ (happ P x) y) ==> (!_0. (!iK xxx. happ (happ _0 iK) xxx = iK) ==> (!iK P. (!y. h4/quantHeuristics/GUESS__EXISTS (happ _0 iK) (happ (happ _1 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS (happ _0 iK) (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c3]: !_1. (!P x. happ (happ _1 P) x <=> (!y. happ (happ P x) y)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__EXISTS__POINT i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__POINT i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c2]: !_1. (!P x. happ (happ _1 P) x <=> (!y. happ (happ P x) y)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__FORALL__GAP i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c1]: !_1. (!P x. happ (happ _1 P) x <=> (!y. happ (happ P x) y)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__FORALL i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__FORALL i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c0]: !_1. (!P x. happ (happ _1 P) x <=> (!y. happ (happ P x) y)) ==> (!_0. (!P y x. happ (happ (happ _0 P) y) x <=> happ (happ P x) y) ==> (!i P. (!y. h4/quantHeuristics/GUESS__FORALL__POINT i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT i (happ _1 P)))
% Goal: !_0. (!i x. happ (happ _0 i) x = happ i (h4/pair/_2C x h4/one/one0)) ==> (!vt i. h4/quantHeuristics/GUESS__EXISTS__GAP i vt <=> h4/quantHeuristics/GUESS__EXISTS__GAP (happ _0 i) vt)
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_Q1098392,TV_Q1098388]: ![V_f, V_g]: (![V_x]: s(TV_Q1098388,happ(s(t_fun(TV_Q1098392,TV_Q1098388),V_f),s(TV_Q1098392,V_x))) = s(TV_Q1098388,happ(s(t_fun(TV_Q1098392,TV_Q1098388),V_g),s(TV_Q1098392,V_x))) => s(t_fun(TV_Q1098392,TV_Q1098388),V_f) = s(t_fun(TV_Q1098392,TV_Q1098388),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_i, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_ones_one,h4s_ones_one0))))) => ![V_vt, V_i]: s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_vt))) = s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(t_fun(TV_u_27b,t_bool),V_vt))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_i, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_ones_one,h4s_ones_one0))))) => ![V_vt, V_i]: s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_vt))) = s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(t_fun(TV_u_27b,t_bool),V_vt))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_i, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_ones_one,h4s_ones_one0))))) => ![V_vt, V_i]: s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_vt))) = s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(t_fun(TV_u_27b,t_bool),V_vt))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_i, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_ones_one,h4s_ones_one0))))) => ![V_vt, V_i]: s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_vt))) = s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(t_fun(TV_u_27b,t_bool),V_vt))))).
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_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_ones_oneu_u_induction, axiom, ![V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,h4s_ones_one0)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,V_x)))))).
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_quantHeuristicss_GUESSu_u_REWRITESu_c4, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v)))) => ?[V_fv]: s(TV_u_27b,V_v) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))).
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_quantHeuristicss_GUESSu_u_REWRITESu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_fv]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_REWRITESu_c5, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v))))) => ?[V_fv]: s(TV_u_27b,V_v) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))).
fof(ah4s_quantHeuristicss_GUESSu_u_REWRITESu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_fv]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_REWRITESu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v))))) => ?[V_fv]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_REWRITESu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v)))) => ?[V_fv]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv))))))))).
fof(ah4s_pairs_EXISTSu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (?[V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))) <=> ?[V_pu_u_1, V_pu_u_2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
fof(ah4s_pairs_FORALLu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))) <=> ![V_pu_u_1, V_pu_u_2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
fof(ah4s_ones_one1, axiom, ![V_v]: s(t_h4s_ones_one,V_v) = s(t_h4s_ones_one,h4s_ones_one0)).
fof(ah4s_ones_oneu_u_axiom, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_h4s_ones_one),V_f) = s(t_fun(TV_u_27a,t_h4s_ones_one),V_g)).
fof(ah4s_bools_boolAxiom, axiom, ![TV_u_27a]: ![V_t2, V_t1]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_fn),s(t_bool,t))) = s(TV_u_27a,V_t1) & s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_fn),s(t_bool,f))) = s(TV_u_27a,V_t2))).
fof(ah4s_ones_oneu_u_Axiom, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_e, V_fn]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool)),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn)))) <=> s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn),s(t_h4s_ones_one,h4s_ones_one0))) = s(TV_u_27a,V_e)) => ![V_e]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool)),V_uu_0),s(TV_u_27a,V_e)))))))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_bools_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_options_OPTIONu_u_GUARDu_u_defu_c0, axiom, s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,t))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0)))).
fof(ah4s_ones_oneu_u_caseu_u_thm, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_ones_oneu_u_case(s(t_h4s_ones_one,h4s_ones_one0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_ones_oneu_u_primu_u_rec, axiom, ![TV_u_27a]: ![V_e]: ?[V_fn]: s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn),s(t_h4s_ones_one,h4s_ones_one0))) = s(TV_u_27a,V_e)).
fof(ah4s_ones_FORALLu_u_ONE, axiom, ![V_P]: (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,V_x)))) <=> p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_EQu_u_THMu_c0, axiom, ![V_b]: (s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,V_b))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0))) <=> p(s(t_bool,V_b)))).
fof(ah4s_ones_oneu_u_DEF, axiom, ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0),s(t_h4s_ones_one,V_x))) = s(t_bool,t) => s(t_h4s_ones_one,h4s_ones_one0) = s(t_h4s_ones_one,h4s_mins_u_40(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0))))).
fof(ah4s_options_NONEu_u_DEF, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_COND, axiom, ![V_b]: s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,V_b))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_b),s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0))),s(t_h4s_options_option(t_h4s_ones_one),h4s_options_none)))).
fof(ah4s_quantHeuristicss_GUESSu_u_EXISTSu_u_GAPu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v)))) => ?[V_fv]: s(TV_u_27b,V_v) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_defu_c1, axiom, s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,f))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_none)).
fof(ah4s_ones_oneu_u_caseu_u_def, axiom, ![TV_u_27a]: ![V_x, V_u]: s(TV_u_27a,h4s_ones_oneu_u_case(s(t_h4s_ones_one,V_u),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x))) => ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x)))))) => ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P)))))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x))) => ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x)))))) => ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P)))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_u_1u_c3, axiom, ![TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_uu_0]: (![V_i, V_y, V_xxx]: s(TV_u_27c,happ(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_h4s_ones_one,V_xxx))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_EQUATIONu_u_EXISTSu_u_GAP, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_i, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_1),s(TV_u_27a,V_i))),s(TV_u_27a,V_x)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_i)) => ![V_uu_0]: (![V_i, V_xxx]: s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27a)),V_uu_0),s(TV_u_27a,V_i))),s(t_h4s_ones_one,V_xxx))) = s(TV_u_27a,V_i) => ![V_i]: p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(t_h4s_ones_one,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27a)),V_uu_0),s(TV_u_27a,V_i))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_1),s(TV_u_27a,V_i))))))))).
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_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_u_u_u_u_u_NEWu_u_FVu_u_1u_c2, axiom, ![TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_uu_0]: (![V_i, V_y, V_xxx]: s(TV_u_27c,happ(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_h4s_ones_one,V_xxx))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_BOOLu_c2, axiom, ![V_uu_1]: (![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_uu_1),s(t_bool,V_x))) = s(t_bool,V_x) => ![V_uu_0]: (![V_ARB]: s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0),s(t_h4s_ones_one,V_ARB))) = s(t_bool,t) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0),s(t_fun(t_bool,t_bool),V_uu_1))))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_c3, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_iy, V_fv]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))) = 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_iy),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_iy, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(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_iy),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_ONEu_u_CASEu_u_u_u_u_u_EXISTSu_u_GAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_Q, V_P]: (![V_x]: ?[V_fv]: s(TV_u_27a,V_x) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_Q),s(TV_u_27b,V_fv))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27b,TV_u_27a),V_Q),s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c5, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x))) => ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x)))))) => ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P)))))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x))) => ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x)))))) => ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P)))))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_UEXISTSu_u_THM2, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_i, V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_uu_0),s(TV_u_27a,V_i))),s(TV_u_27b,V_x))) = s(TV_u_27a,V_i) => ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_uu_0),s(TV_u_27a,V_i))),s(t_fun(TV_u_27a,t_bool),V_P)))) => s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_i)))))).
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_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_WEAKENu_u_THMu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_u_u_u_u_u_NEWu_u_FVu_c2, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_iy, V_fv]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))) = 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_iy),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_iy, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(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_iy),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
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_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_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_quantHeuristicss_GUESSu_u_RULESu_u_EQUIVu_c6, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P1, V_P2, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P1))),s(t_fun(TV_u_27b,t_bool),V_P2))),s(TV_u_27b,V_x)))) <=> s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P1),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P2),s(TV_u_27b,V_x)))) => ![V_uu_0]: (![V_P1, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P1))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P1),s(TV_u_27b,V_x))) => ![V_i, V_P2, V_P1]: ((p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P1)))))) & p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P2))))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P1))),s(t_fun(TV_u_27b,t_bool),V_P2)))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_EQUIVu_c7, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P1, V_P2, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P1))),s(t_fun(TV_u_27b,t_bool),V_P2))),s(TV_u_27b,V_x)))) <=> s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P1),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P2),s(TV_u_27b,V_x)))) => ![V_uu_0]: (![V_P1, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P1))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P1),s(TV_u_27b,V_x))) => ![V_i, V_P2, V_P1]: ((p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P1)))))) & p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P2))))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P1))),s(t_fun(TV_u_27b,t_bool),V_P2)))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_EQUIVu_c5, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P1, V_P2, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P1))),s(t_fun(TV_u_27b,t_bool),V_P2))),s(TV_u_27b,V_x)))) <=> s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P1),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P2),s(TV_u_27b,V_x)))) => ![V_uu_0]: (![V_P1, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P1))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P1),s(TV_u_27b,V_x))) => ![V_i, V_P2, V_P1]: ((p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P1)))))) & p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P2))))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P1))),s(t_fun(TV_u_27b,t_bool),V_P2)))))))))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_UEXISTSu_u_THM4, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_i, V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27a)),V_uu_1),s(TV_u_27a,V_i))),s(TV_u_27c,V_x))) = s(TV_u_27a,V_i) => ![V_uu_0]: (![V_i, V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_uu_0),s(TV_u_27a,V_i))),s(TV_u_27b,V_x))) = s(TV_u_27a,V_i) => ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_uu_0),s(TV_u_27a,V_i))),s(t_fun(TV_u_27a,t_bool),V_P)))) => (p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27a)),V_uu_1),s(TV_u_27a,V_i))),s(t_fun(TV_u_27a,t_bool),V_P)))) => s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,t)))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_NEGu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x)))))) => ![V_uu_0]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x))) => ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P)))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_c2, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P)))))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_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_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,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_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c4, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c5, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_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_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_u_1u_c2, axiom, ![TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_uu_0]: (![V_i, V_y, V_xxx]: s(TV_u_27c,happ(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_h4s_ones_one,V_xxx))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_u_1u_c1, axiom, ![TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_uu_0]: (![V_i, V_y, V_xxx]: s(TV_u_27c,happ(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_h4s_ones_one,V_xxx))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_u_1u_c0, axiom, ![TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_uu_0]: (![V_i, V_y, V_xxx]: s(TV_u_27c,happ(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_h4s_ones_one,V_xxx))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_i))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_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_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_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_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_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_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_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_c2, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_iy, V_fv]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))) = 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_iy),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_iy, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(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_iy),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_iy, V_fv]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))) = 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_iy),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_iy, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(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_iy),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_u_u_u_u_u_NEWu_u_FVu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_iy, V_fv]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))) = 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_iy),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_fv))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_iy, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(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_iy),s(TV_u_27a,V_y))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_iy))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c4, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x))) => ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x)))))) => ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P)))))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x))) => ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27b,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_x)))))) => ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P)))))))).
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_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_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_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_THM, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))) => ![V_P]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),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, V_y]: ((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_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
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_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(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_27b,V_y)).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c5, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P)))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c4, axiom, ![TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_uu_0]: (![V_iK, V_xxx]: s(TV_u_27c,happ(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27c,t_fun(t_h4s_ones_one,TV_u_27c)),V_uu_0),s(TV_u_27c,V_iK))),s(t_h4s_ones_one,V_xxx))) = s(TV_u_27c,V_iK) => ![V_iK, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27c,t_fun(t_h4s_ones_one,TV_u_27c)),V_uu_0),s(TV_u_27c,V_iK))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(t_h4s_ones_one,TV_u_27c),happ(s(t_fun(TV_u_27c,t_fun(t_h4s_ones_one,TV_u_27c)),V_uu_0),s(TV_u_27c,V_iK))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c3, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P)))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c2, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P)))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P)))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_FORALLu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_1]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27c,V_x)))) <=> ![V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))) => ![V_uu_0]: (![V_P, V_y, V_x]: s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y))),s(TV_u_27c,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))) => ![V_i, V_P]: (![V_y]: p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27b,TV_u_27c),V_i),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_bool)),V_uu_1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_P)))))))))).
fof(ch4s_quantHeuristicss_GUESSu_u_RULESu_u_ELIMu_u_UNITu_c4, conjecture, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_i, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_ones_one,h4s_ones_one0))))) => ![V_vt, V_i]: s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_vt))) = s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_ones_one),TV_u_27b),V_i))),s(t_fun(TV_u_27b,t_bool),V_vt))))).
