%   ORIGINAL: h4/quantHeuristics/GUESS__RULES__CONSTANT__EXISTS
% 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/quantHeuristics/GUESS__EXISTS__FORALL__REWRITES_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. P v ==> (?fv. P (i fv)))
% 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/quantHeuristics/GUESSES__NEG__DUALITY_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL i P
% Assm: h4/quantHeuristics/GUESS__REWRITES_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. P v ==> (?fv. P (i fv)))
% Assm: h4/quantHeuristics/GUESSES__NEG__REWRITE_c1: !i P. h4/quantHeuristics/GUESS__FORALL i (\x. ~P x) <=> h4/quantHeuristics/GUESS__EXISTS i (\x. P x)
% Assm: h4/quantHeuristics/GUESSES__NEG__REWRITE_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i (\x. ~P x) <=> h4/quantHeuristics/GUESS__FORALL i (\x. P x)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/quantHeuristics/GUESS__EXISTS__FORALL__REWRITES_c1: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~P v ==> (?fv. ~P (i fv)))
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/quantHeuristics/GUESS__EXISTS__def: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (?v. P v) <=> (?fv. P (i fv))
% Assm: h4/quantHeuristics/GUESSES__WEAKEN__THM_c2: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P ==> h4/quantHeuristics/GUESS__EXISTS i P
% Assm: h4/quantHeuristics/GUESSES__WEAKEN__THM_c3: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P ==> h4/quantHeuristics/GUESS__EXISTS i P
% Assm: h4/quantHeuristics/GUESS__RULES__NEG_c3: !i P. h4/quantHeuristics/GUESS__FORALL i (\x. P x) ==> h4/quantHeuristics/GUESS__EXISTS i (\x. ~P x)
% Assm: h4/quantHeuristics/GUESS__RULES__NEG_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i (\x. P x) ==> h4/quantHeuristics/GUESS__FORALL i (\x. ~P x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/quantHeuristics/GUESS__EXISTS__THM: !i P. h4/quantHeuristics/GUESS__EXISTS i P ==> ($exists P <=> (?fv. P (i fv)))
% Assm: h4/quantHeuristics/GUESSES__UEXISTS__THM1: !i P. h4/quantHeuristics/GUESS__EXISTS (\x. i) P ==> (h4/bool/_3F_21 P <=> P i /\ (!v. P v ==> v = i))
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% 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/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_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_c4: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__POINT i P
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/T__DEF: T <=> (\x. x) = (\x. x)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% 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_c3: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~P (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_c4: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P <=> (!v. P v ==> (?fv. v = 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/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 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/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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_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_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/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/numeral/bit__initiality: !zf b2f b1f. ?f. f h4/arithmetic/ZERO = zf /\ (!n. f (h4/arithmetic/BIT1 n) = b1f n (f n)) /\ (!n. f (h4/arithmetic/BIT2 n) = b2f n (f n))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/EXISTS__THM: !f. $exists f <=> (?x. f x)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/quantHeuristics/GUESS__POINT__THM_c0: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P ==> ((?v. P v) <=> T)
% Assm: h4/quantHeuristics/GUESS__EXISTS__POINT__THM: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P ==> ($exists P <=> T)
% Assm: h4/quantHeuristics/GUESS__FORALL__def: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. P v) <=> (!fv. P (i fv))
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% 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/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/COND__DEF: h4/bool/COND = (\t t1 t2. h4/min/_40 (\x. ((t <=> T) ==> x = t1) /\ ((t <=> F) ==> x = t2)))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/FORALL__BOOL: !P. (!b. P b) <=> P T /\ P F
% Assm: h4/bool/bool__INDUCT: !P. P T /\ P F ==> (!b. P b)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/COND__ABS: !g f b. (\x. h4/bool/COND b (f x) (g x)) = h4/bool/COND b f g
% Assm: h4/bool/NOT__DEF: $not = (\t. t ==> F)
% Assm: h4/bool/BOUNDED__DEF: h4/bool/BOUNDED = (\v. T)
% Assm: h4/ConseqConv/IMP__CLAUSES__XT: !t. t ==> T <=> T
% Assm: h4/quantHeuristics/GUESS__EXISTS__POINT__def: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P <=> (!fv. P (i fv))
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/ConseqConv/IMP__CLAUSES__TX: !t. T ==> t <=> t
% Assm: h4/quantHeuristics/GUESS__FORALL__POINT__def: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~P (i fv))
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/ConseqConv/AND__CLAUSES__XT: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/ConseqConv/COND__CLAUSES__FT: !x c. h4/bool/COND c x T <=> c ==> x
% Assm: h4/sat/EQT__Imp1: !b. b ==> (b <=> T)
% Assm: h4/bool/BOOL__FUN__CASES__THM: !f. f = (\b. T) \/ f = (\b. F) \/ f = (\b. b) \/ f = (\b. ~b)
% Assm: h4/ConseqConv/IMP__CLAUSES__FX: !t. F ==> t <=> T
% Assm: h4/ConseqConv/OR__CLAUSES__XT: !t. 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/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/ConseqConv/OR__CLAUSES__TX: !t. T \/ t <=> T
% Assm: h4/ConseqConv/IMP__CLAUSES__XX: !t. t ==> t <=> T
% Assm: h4/ConseqConv/COND__CLAUSES__TT: !x c. h4/bool/COND c T x <=> ~c ==> x
% Assm: h4/numeral/iSUB__DEF_c1: !x n b. h4/numeral/iSUB b (h4/arithmetic/BIT1 n) x = h4/bool/COND b (h4/numeral/iBIT__cases x (h4/arithmetic/BIT1 n) (\m. h4/numeral/iDUB (h4/numeral/iSUB T n m)) (\m. h4/arithmetic/BIT1 (h4/numeral/iSUB F n m))) (h4/numeral/iBIT__cases x (h4/numeral/iDUB n) (\m. h4/arithmetic/BIT1 (h4/numeral/iSUB F n m)) (\m. h4/numeral/iDUB (h4/numeral/iSUB F n m)))
% Assm: h4/numeral/iSUB__DEF_c2: !x n b. h4/numeral/iSUB b (h4/arithmetic/BIT2 n) x = h4/bool/COND b (h4/numeral/iBIT__cases x (h4/arithmetic/BIT2 n) (\m. h4/arithmetic/BIT1 (h4/numeral/iSUB T n m)) (\m. h4/numeral/iDUB (h4/numeral/iSUB T n m))) (h4/numeral/iBIT__cases x (h4/arithmetic/BIT1 n) (\m. h4/numeral/iDUB (h4/numeral/iSUB T n m)) (\m. h4/arithmetic/BIT1 (h4/numeral/iSUB F n m)))
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/ConseqConv/AND__CLAUSES__TX: !t. T /\ t <=> t
% Assm: h4/bool/BOOL__FUN__INDUCT: !P. P (\b. T) /\ P (\b. F) /\ P (\b. b) /\ P (\b. ~b) ==> (!f. P f)
% Assm: h4/quantHeuristics/GUESS__RULES__BOOL_c2: h4/quantHeuristics/GUESS__EXISTS__GAP (\ARB. T) (\x. x)
% Assm: h4/bool/AND1__THM: !t2 t1. t1 /\ t2 ==> t1
% Assm: h4/bool/EXCLUDED__MIDDLE0: !t. t \/ ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Goal: !p i. h4/quantHeuristics/GUESS__EXISTS i (\x. p) <=> T
%   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_quantHeuristicss_GUESSu_u_EXISTSu_u_FORALLu_u_REWRITESu_c0]: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. happ P v ==> (?fv. happ P (happ i fv)))
% 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_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_GUESSu_u_REWRITESu_c0]: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. happ P v ==> (?fv. happ P (happ i fv)))
% 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_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_TRUTH]: T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_FORALLu_u_REWRITESu_c1]: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~happ P v ==> (?fv. ~happ P (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_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_def]: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (?v. happ P v) <=> (?fv. happ P (happ i fv))
% Assm [h4s_quantHeuristicss_GUESSESu_u_WEAKENu_u_THMu_c2]: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P ==> h4/quantHeuristics/GUESS__EXISTS i P
% 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_NEGu_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 i (happ _0 P) ==> h4/quantHeuristics/GUESS__EXISTS i (happ _0 (happ _1 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_NEGu_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 _0 (happ _1 P))))
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_THM]: !i P. h4/quantHeuristics/GUESS__EXISTS i P ==> ($exists P <=> (?fv. happ P (happ i fv)))
% Assm [h4s_quantHeuristicss_GUESSESu_u_UEXISTSu_u_THM1]: !_0. (!i x. happ (happ _0 i) x = i) ==> (!i P. h4/quantHeuristics/GUESS__EXISTS (happ _0 i) P ==> (h4/bool/_3F_21 P <=> happ P i /\ (!v. happ P v ==> v = i)))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% 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_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_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_c4]: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__POINT i P
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_Tu_u_DEF]: T <=> (!x. x <=> x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% 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_c3]: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~happ P (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_c4]: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P <=> (!v. happ P v ==> (?fv. v = 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_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% 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_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~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_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_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_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_numerals_bitu_u_initiality]: !zf b2f b1f. ?f. happ f h4/arithmetic/ZERO = zf /\ (!n. happ f (h4/arithmetic/BIT1 n) = happ (happ b1f n) (happ f n)) /\ (!n. happ f (h4/arithmetic/BIT2 n) = happ (happ b2f n) (happ f n))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_EXISTSu_u_THM]: !f. $exists f <=> (?x. happ f x)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_quantHeuristicss_GUESSu_u_POINTu_u_THMu_c0]: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P ==> ((?v. happ P v) <=> T)
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_POINTu_u_THM]: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P ==> ($exists P <=> T)
% Assm [h4s_quantHeuristicss_GUESSu_u_FORALLu_u_def]: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. happ P v) <=> (!fv. happ P (happ i fv))
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% 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_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_CONDu_u_DEF]: !_0. (!x x x' x''. happ (happ (happ (happ _0 x) x) x') x'' <=> ((x <=> T) ==> x'' = x) /\ ((x <=> F) ==> x'' = x')) ==> (!x x x'. h4/bool/COND x x x' = h4/min/_40 (happ (happ (happ _0 x) x) x'))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_FORALLu_u_BOOL]: !P. (!b. happ P b) <=> happ P T /\ happ P F
% Assm [h4s_bools_boolu_u_INDUCT]: !P. happ P T /\ happ P F ==> (!b. happ P b)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_CONDu_u_ABS]: !g f b x. h4/bool/COND b (happ f x) (happ g x) = happ (h4/bool/COND b f g) x
% Assm [h4s_bools_NOTu_u_DEF]: !x. happ $not x <=> x ==> F
% Assm [h4s_bools_BOUNDEDu_u_DEF]: !x. h4/bool/BOUNDED x <=> T
% Assm [h4s_ConseqConvs_IMPu_u_CLAUSESu_u_XT]: !t. t ==> T <=> T
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_POINTu_u_def]: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P <=> (!fv. happ P (happ i fv))
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_ConseqConvs_IMPu_u_CLAUSESu_u_TX]: !t. T ==> t <=> t
% Assm [h4s_quantHeuristicss_GUESSu_u_FORALLu_u_POINTu_u_def]: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~happ P (happ i fv))
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_ConseqConvs_ANDu_u_CLAUSESu_u_XT]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_ConseqConvs_CONDu_u_CLAUSESu_u_FT]: !x c. h4/bool/COND c x T <=> c ==> x
% Assm [h4s_sats_EQTu_u_Imp1]: !b. b ==> (b <=> T)
% Assm [h4s_bools_BOOLu_u_FUNu_u_CASESu_u_THM]: !f. (!x. happ f x <=> T) \/ (!x. happ f x <=> F) \/ (!x. happ f x <=> x) \/ (!x. happ f x <=> ~x)
% Assm [h4s_ConseqConvs_IMPu_u_CLAUSESu_u_FX]: !t. F ==> t <=> T
% Assm [h4s_ConseqConvs_ORu_u_CLAUSESu_u_XT]: !t. 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_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_ConseqConvs_ORu_u_CLAUSESu_u_TX]: !t. T \/ t <=> T
% Assm [h4s_ConseqConvs_IMPu_u_CLAUSESu_u_XX]: !t. t ==> t <=> T
% Assm [h4s_ConseqConvs_CONDu_u_CLAUSESu_u_TT]: !x c. h4/bool/COND c T x <=> ~c ==> x
% Assm [h4s_numerals_iSUBu_u_DEFu_c1]: !_2. (!n m. happ (happ _2 n) m = h4/numeral/iDUB (h4/numeral/iSUB F n m)) ==> (!_1. (!n m. happ (happ _1 n) m = h4/arithmetic/BIT1 (h4/numeral/iSUB F n m)) ==> (!_0. (!n m. happ (happ _0 n) m = h4/numeral/iDUB (h4/numeral/iSUB T n m)) ==> (!x n b. h4/numeral/iSUB b (h4/arithmetic/BIT1 n) x = h4/bool/COND b (h4/numeral/iBIT__cases x (h4/arithmetic/BIT1 n) (happ _0 n) (happ _1 n)) (h4/numeral/iBIT__cases x (h4/numeral/iDUB n) (happ _1 n) (happ _2 n)))))
% Assm [h4s_numerals_iSUBu_u_DEFu_c2]: !_2. (!n m. happ (happ _2 n) m = h4/arithmetic/BIT1 (h4/numeral/iSUB F n m)) ==> (!_1. (!n m. happ (happ _1 n) m = h4/numeral/iDUB (h4/numeral/iSUB T n m)) ==> (!_0. (!n m. happ (happ _0 n) m = h4/arithmetic/BIT1 (h4/numeral/iSUB T n m)) ==> (!x n b. h4/numeral/iSUB b (h4/arithmetic/BIT2 n) x = h4/bool/COND b (h4/numeral/iBIT__cases x (h4/arithmetic/BIT2 n) (happ _0 n) (happ _1 n)) (h4/numeral/iBIT__cases x (h4/arithmetic/BIT1 n) (happ _1 n) (happ _2 n)))))
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_ConseqConvs_ANDu_u_CLAUSESu_u_TX]: !t. T /\ t <=> t
% Assm [h4s_bools_BOOLu_u_FUNu_u_INDUCT]: !_3. (!b. happ _3 b <=> ~b) ==> (!_2. (!b. happ _2 b <=> b) ==> (!_1. (!b. happ _1 b <=> F) ==> (!_0. (!b. happ _0 b <=> T) ==> (!P. happ P _0 /\ happ P _1 /\ happ P _2 /\ happ P _3 ==> (!f. happ P f)))))
% 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_AND1u_u_THM]: !t2 t1. t1 /\ t2 ==> t1
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE0]: !t. t \/ ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Goal: !_0. (!p x. happ (happ _0 p) x <=> p) ==> (!p i. h4/quantHeuristics/GUESS__EXISTS i (happ _0 p) <=> T)
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_Q1096176,TV_Q1096172]: ![V_f, V_g]: (![V_x]: s(TV_Q1096172,happ(s(t_fun(TV_Q1096176,TV_Q1096172),V_f),s(TV_Q1096176,V_x))) = s(TV_Q1096172,happ(s(t_fun(TV_Q1096176,TV_Q1096172),V_g),s(TV_Q1096176,V_x))) => s(t_fun(TV_Q1096176,TV_Q1096172),V_f) = s(t_fun(TV_Q1096176,TV_Q1096172),V_g))).
fof(ah4s_quantHeuristicss_GUESSu_u_EXISTSu_u_FORALLu_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_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_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_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_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_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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_quantHeuristicss_GUESSu_u_EXISTSu_u_FORALLu_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_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_EXISTSu_u_def, 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_quantHeuristicss_GUESSESu_u_WEAKENu_u_THMu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![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)))) => 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_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_NEGu_c3, 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_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)))))) => 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),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_NEGu_c0, 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_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)))))) => 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),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_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_quantHeuristicss_GUESSu_u_EXISTSu_u_THM, 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)))) => (p(s(t_bool,d_exists(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_GUESSESu_u_UEXISTSu_u_THM1, 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_exists(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_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_P)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_i)))) & ![V_v]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_v)))) => s(TV_u_27a,V_v) = s(TV_u_27a,V_i))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_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_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_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_Tu_u_DEF, axiom, (p(s(t_bool,t)) <=> ![V_x]: s(t_bool,V_x) = s(t_bool,V_x))).
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_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_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_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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> 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_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_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_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_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_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_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_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_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_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_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_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_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_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_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_numerals_bitu_u_initiality, axiom, ![TV_u_27a]: ![V_zf, V_b2f, V_b1f]: ?[V_f]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(TV_u_27a,V_zf) & (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_b1f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_b2f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_sats_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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),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_GUESSu_u_POINTu_u_THMu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![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_v]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v)))) <=> p(s(t_bool,t))))).
fof(ah4s_quantHeuristicss_GUESSu_u_EXISTSu_u_POINTu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![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)))) => s(t_bool,d_exists(s(t_fun(TV_u_27b,t_bool),V_P))) = s(t_bool,t))).
fof(ah4s_quantHeuristicss_GUESSu_u_FORALLu_u_def, 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_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_CONDu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_x0, V_xi_, V_xi_i_]: (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)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(t_bool,V_x0))),s(TV_u_27a,V_xi_))),s(TV_u_27a,V_xi_i_)))) <=> ((s(t_bool,V_x0) = s(t_bool,t) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_x)) & (s(t_bool,V_x0) = s(t_bool,f) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_xi_)))) => ![V_x, V_x0, V_xi_]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_x),s(TV_u_27a,V_x0),s(TV_u_27a,V_xi_))) = s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x0))),s(t_bool,V_x))),s(TV_u_27a,V_xi_))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_BOOL, axiom, ![V_P]: (![V_b]: p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_P),s(t_bool,V_b)))) <=> (p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_P),s(t_bool,t)))) & p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_P),s(t_bool,f))))))).
fof(ah4s_bools_boolu_u_INDUCT, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_P),s(t_bool,t)))) & p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_P),s(t_bool,f))))) => ![V_b]: p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_P),s(t_bool,V_b)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_CONDu_u_ABS, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_b, V_x]: s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x)))).
fof(ah4s_bools_NOTu_u_DEF, axiom, ![V_x]: (p(s(t_bool,happ(s(t_fun(t_bool,t_bool),d_not),s(t_bool,V_x)))) <=> (p(s(t_bool,V_x)) => p(s(t_bool,f))))).
fof(ah4s_bools_BOUNDEDu_u_DEF, axiom, ![V_x]: s(t_bool,h4s_bools_bounded(s(t_bool,V_x))) = s(t_bool,t)).
fof(ah4s_ConseqConvs_IMPu_u_CLAUSESu_u_XT, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_quantHeuristicss_GUESSu_u_EXISTSu_u_POINTu_u_def, 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_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_ConseqConvs_IMPu_u_CLAUSESu_u_TX, 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_FORALLu_u_POINTu_u_def, 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_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_ConseqConvs_ANDu_u_CLAUSESu_u_XT, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_ConseqConvs_CONDu_u_CLAUSESu_u_FT, axiom, ![V_x, V_c]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_c),s(t_bool,V_x),s(t_bool,t)))) <=> (p(s(t_bool,V_c)) => p(s(t_bool,V_x))))).
fof(ah4s_sats_EQTu_u_Imp1, axiom, ![V_b]: (p(s(t_bool,V_b)) => s(t_bool,V_b) = s(t_bool,t))).
fof(ah4s_bools_BOOLu_u_FUNu_u_CASESu_u_THM, axiom, ![V_f]: (![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_f),s(t_bool,V_x))) = s(t_bool,t) | (![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_f),s(t_bool,V_x))) = s(t_bool,f) | (![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_f),s(t_bool,V_x))) = s(t_bool,V_x) | ![V_x]: (p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_f),s(t_bool,V_x)))) <=> ~ (p(s(t_bool,V_x)))))))).
fof(ah4s_ConseqConvs_IMPu_u_CLAUSESu_u_FX, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_ConseqConvs_ORu_u_CLAUSESu_u_XT, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,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_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_ConseqConvs_ORu_u_CLAUSESu_u_TX, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_ConseqConvs_IMPu_u_CLAUSESu_u_XX, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_ConseqConvs_CONDu_u_CLAUSESu_u_TT, axiom, ![V_x, V_c]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_c),s(t_bool,t),s(t_bool,V_x)))) <=> (~ (p(s(t_bool,V_c))) => p(s(t_bool,V_x))))).
fof(ah4s_numerals_iSUBu_u_DEFu_c1, axiom, ![V_uu_2]: (![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_2),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,f),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) => ![V_uu_1]: (![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,f),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) => ![V_uu_0]: (![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) => ![V_x, V_n, V_b]: s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,V_b),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_b),s(t_h4s_nums_num,h4s_numerals_ibitu_u_cases(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_numerals_ibitu_u_cases(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_2),s(t_h4s_nums_num,V_n))))))))))).
fof(ah4s_numerals_iSUBu_u_DEFu_c2, axiom, ![V_uu_2]: (![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_2),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,f),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) => ![V_uu_1]: (![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) => ![V_uu_0]: (![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) => ![V_x, V_n, V_b]: s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,V_b),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_b),s(t_h4s_nums_num,h4s_numerals_ibitu_u_cases(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_numerals_ibitu_u_cases(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_2),s(t_h4s_nums_num,V_n))))))))))).
fof(ah4s_bools_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t)).
fof(ah4s_ConseqConvs_ANDu_u_CLAUSESu_u_TX, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_BOOLu_u_FUNu_u_INDUCT, axiom, ![V_uu_3]: (![V_b]: (p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_uu_3),s(t_bool,V_b)))) <=> ~ (p(s(t_bool,V_b)))) => ![V_uu_2]: (![V_b]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_uu_2),s(t_bool,V_b))) = s(t_bool,V_b) => ![V_uu_1]: (![V_b]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_uu_1),s(t_bool,V_b))) = s(t_bool,f) => ![V_uu_0]: (![V_b]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_uu_0),s(t_bool,V_b))) = s(t_bool,t) => ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(t_bool,t_bool),t_bool),V_P),s(t_fun(t_bool,t_bool),V_uu_0)))) & (p(s(t_bool,happ(s(t_fun(t_fun(t_bool,t_bool),t_bool),V_P),s(t_fun(t_bool,t_bool),V_uu_1)))) & (p(s(t_bool,happ(s(t_fun(t_fun(t_bool,t_bool),t_bool),V_P),s(t_fun(t_bool,t_bool),V_uu_2)))) & p(s(t_bool,happ(s(t_fun(t_fun(t_bool,t_bool),t_bool),V_P),s(t_fun(t_bool,t_bool),V_uu_3))))))) => ![V_f]: p(s(t_bool,happ(s(t_fun(t_fun(t_bool,t_bool),t_bool),V_P),s(t_fun(t_bool,t_bool),V_f)))))))))).
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_AND1u_u_THM, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t1)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE0, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ch4s_quantHeuristicss_GUESSu_u_RULESu_u_CONSTANTu_u_EXISTS, conjecture, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_p, V_x]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_bool,V_p))),s(TV_u_27b,V_x))) = s(t_bool,V_p) => ![V_p, V_i]: 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_bool,t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_bool,V_p))))) = s(t_bool,t))).
