%   ORIGINAL: h4/quantHeuristics/GUESS__RULES__EXISTS__UNIQUE_c0
% 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__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)
% Assm: h4/quantHeuristics/GUESS__RULES__EXISTS_c3: !i P. (!y. h4/quantHeuristics/GUESS__FORALL__POINT i (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT i (\x. ?y. P x y)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!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/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_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/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_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_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_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_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/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/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/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> 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_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/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/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% 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/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__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% 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/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% 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/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> 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_c1: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~P v ==> (?fv. ~P (i fv)))
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/quantHeuristics/GUESS__REWRITES_c3: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~P (i fv))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/quantHeuristics/GUESS__REWRITES_c2: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i P <=> (!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/PAIR__FUN__THM: !P. h4/bool/_3F_21 (\f. P f) <=> h4/bool/_3F_21 (\p. P (\a. h4/pair/_2C (h4/pair/FST p a) (h4/pair/SND p a)))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% 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/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/quantHeuristics/GUESS__RULES__EXISTS_c5: !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__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/quantHeuristics/GUESS__RULES__EXISTS_c0: !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__EXISTS_c1: !i P. (!y. h4/quantHeuristics/GUESS__EXISTS i (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS i (\x. ?y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__EXISTS______NEW__FV_c1: !iy P. (!y. h4/quantHeuristics/GUESS__EXISTS (iy y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS (\fv. iy (h4/pair/FST fv) (h4/pair/SND fv)) (\x. ?y. P x y)
% Assm: h4/quantHeuristics/GUESS__RULES__EXISTS______NEW__FV_c3: !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__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/relation/O__DEF: !z x R2 R1. h4/relation/O R1 R2 x z <=> (?y. R2 x y /\ R1 y z)
% Assm: h4/quantHeuristics/GUESS__RULES__EXISTS______NEW__FV_c0: !iy P. (!y. h4/quantHeuristics/GUESS__EXISTS__POINT (iy y) (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__POINT (\fv. iy (h4/pair/FST fv) (h4/pair/SND fv)) (\x. ?y. P x y)
% Assm: h4/relation/WF__RECURSION__THM: !R. h4/relation/WF R ==> (!M. h4/bool/_3F_21 (\f. !x. f x = M (h4/relation/RESTRICT f R x) x))
% Assm: h4/quantHeuristics/GUESSES__UEXISTS__THM3: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT (\x. i) P ==> (h4/bool/_3F_21 P <=> (!v. P v ==> v = i))
% 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/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/list/list__Axiom__old: !x f. h4/bool/_3F_21 (\fn1. fn1 h4/list/NIL = x /\ (!h t. fn1 (h4/list/CONS h t) = f (fn1 t) h t))
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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/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/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/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/NOT__IMP: !B A. ~(A ==> B) <=> A /\ ~B
% 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/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/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/ETA__THM: !M. (\x. M x) = M
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> 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/relation/WFREC__THM: !R M. h4/relation/WF R ==> (!x. h4/relation/WFREC R M x = M (h4/relation/RESTRICT (h4/relation/WFREC R M) R x) x)
% Assm: h4/relation/RESTRICT__DEF: !x f R. h4/relation/RESTRICT f R x = (\y. h4/bool/COND (R y x) (f y) h4/bool/ARB)
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/hidden____20__40ind____typelist0____def: h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/list/list__repfns_c1: !r. (\a0_27. !_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_27 a1) ==> _27list_27 a0_270) ==> _27list_27 a0_27) r <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Goal: !i P. (!y. h4/quantHeuristics/GUESS__FORALL__POINT i (\x. P x y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT i (\x. h4/bool/_3F_21 (\y. P x y))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_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)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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__FORALL__POINT i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__FORALL__POINT i (happ _1 P)))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% 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_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_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_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_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_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_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_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_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_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_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_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_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_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_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% 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_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_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% 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_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% 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_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> 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_c1]: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~happ P v ==> (?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_quantHeuristicss_GUESSu_u_REWRITESu_c3]: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~happ P (happ i fv))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_c0]: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. happ P v ==> (?fv. happ P (happ i fv)))
% Assm [h4s_pairs_PAIRu_u_FUNu_u_THM]: !_2. (!p a. happ (happ _2 p) a = h4/pair/_2C (happ (h4/pair/FST p) a) (happ (h4/pair/SND p) a)) ==> (!_1. (!P p. happ (happ _1 P) p <=> happ P (happ _2 p)) ==> (!_0. (!P f. happ (happ _0 P) f <=> happ P f) ==> (!P. h4/bool/_3F_21 (happ _0 P) <=> h4/bool/_3F_21 (happ _1 P))))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% 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_bools_LEFTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. happ P x ==> Q) <=> (?x. happ P x) ==> Q
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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__FORALL__GAP i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__FORALL__GAP i (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_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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__EXISTS__POINT i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__POINT i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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__EXISTS i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS i (happ _1 P)))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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__EXISTS (happ iy y) (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS (happ _1 iy) (happ _2 P))))
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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__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_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_relations_Ou_u_DEF]: !z x R2 R1. h4/relation/O R1 R2 x z <=> (?y. happ (happ R2 x) y /\ happ (happ R1 y) z)
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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__EXISTS__POINT (happ iy y) (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__POINT (happ _1 iy) (happ _2 P))))
% Assm [h4s_relations_WFu_u_RECURSIONu_u_THM]: !_0. (!M R f. happ (happ (happ _0 M) R) f <=> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)) ==> (!R. h4/relation/WF R ==> (!M. h4/bool/_3F_21 (happ (happ _0 M) R)))
% Assm [h4s_quantHeuristicss_GUESSESu_u_UEXISTSu_u_THM3]: !_0. (!i x. happ (happ _0 i) x = i) ==> (!i P. h4/quantHeuristics/GUESS__EXISTS__POINT (happ _0 i) P ==> (h4/bool/_3F_21 P <=> (!v. happ P v ==> v = i)))
% 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_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_lists_listu_u_Axiomu_u_old]: !_0. (!x f fn1. happ (happ (happ _0 x) f) fn1 <=> happ fn1 h4/list/NIL = x /\ (!h t. happ fn1 (happ (happ h4/list/CONS h) t) = happ (happ (happ f (happ fn1 t)) h) t)) ==> (!x f. h4/bool/_3F_21 (happ (happ _0 x) f))
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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_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_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_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% 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_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_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !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_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_combins_ou_u_THM]: !x g f. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_ETAu_u_THM]: !M x. happ M x = happ M x
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> 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_relations_WFRECu_u_THM]: !R M. h4/relation/WF R ==> (!x. happ (h4/relation/WFREC R M) x = happ (happ M (h4/relation/RESTRICT (h4/relation/WFREC R M) R x)) x)
% Assm [h4s_relations_RESTRICTu_u_DEF]: !x f R x'. happ (h4/relation/RESTRICT f R x) x' = h4/bool/COND (happ (happ R x') x) (happ f x') h4/bool/ARB
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_lists_listu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 r) <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r)
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Goal: !_2. (!P x y. happ (happ (happ _2 P) x) y <=> happ (happ P x) y) ==> (!_1. (!P x. happ (happ _1 P) x <=> h4/bool/_3F_21 (happ (happ _2 P) x)) ==> (!_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))))
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_Q1098242,TV_Q1098238]: ![V_f, V_g]: (![V_x]: s(TV_Q1098238,happ(s(t_fun(TV_Q1098242,TV_Q1098238),V_f),s(TV_Q1098242,V_x))) = s(TV_Q1098238,happ(s(t_fun(TV_Q1098242,TV_Q1098238),V_g),s(TV_Q1098242,V_x))) => s(t_fun(TV_Q1098242,TV_Q1098238),V_f) = s(t_fun(TV_Q1098242,TV_Q1098238),V_g))).
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(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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_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(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,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_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_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_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_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_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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_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_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_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_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> 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_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_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_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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_PAIRu_u_FUNu_u_THM, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_2]: (![V_p, V_a]: s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),V_p))),s(TV_u_27a,V_a))) = s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_pairs_fst(s(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),V_p))),s(TV_u_27a,V_a))),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),h4s_pairs_snd(s(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),V_p))),s(TV_u_27a,V_a))))) => ![V_uu_1]: (![V_P, V_p]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),t_bool)),V_uu_1),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_P))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),V_p))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),V_p))))) => ![V_uu_0]: (![V_P, V_f]: s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool)),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_P))),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),V_f))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),V_f))) => ![V_P]: s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool)),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_P))))) = s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),t_bool)),V_uu_1),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_P))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
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_bools_LEFTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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_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_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_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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_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_EXISTSu_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_exists(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_exists(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_EXISTSu_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_exists(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_exists(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_EXISTSu_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_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_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_relations_Ou_u_DEF, axiom, ![TV_u_27g,TV_u_27h,TV_u_27k]: ![V_z, V_x, V_R2, V_R1]: (p(s(t_bool,h4s_relations_o(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2),s(TV_u_27g,V_x),s(TV_u_27k,V_z)))) <=> ?[V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27h,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2),s(TV_u_27g,V_x))),s(TV_u_27h,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27k,t_bool),happ(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(TV_u_27h,V_y))),s(TV_u_27k,V_z))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_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_existsu_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_existsu_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_relations_WFu_u_RECURSIONu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_M, V_R, V_f]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool))),V_uu_0),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))) => ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_M]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool))),V_uu_0),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_UEXISTSu_u_THM3, 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_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_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_P)))) <=> ![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_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_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_lists_listu_u_Axiomu_u_old, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_x, V_f, V_fn1]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool))),V_uu_0),s(TV_u_27b,V_x))),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),V_f))),s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn1)))) <=> (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_x) & ![V_h, V_t]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn1),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),V_f),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn1),s(t_h4s_lists_list(TV_u_27a),V_t))))),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) => ![V_x, V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),t_bool))),V_uu_0),s(TV_u_27b,V_x))),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b))),V_f)))))))).
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_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_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_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_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_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_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_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_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_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_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_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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(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_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_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
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_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(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_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_ETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_M, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_M),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_M),s(TV_u_27a,V_x)))).
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_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_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_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_relations_WFRECu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_R, V_M]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x))))).
fof(ah4s_relations_RESTRICTu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_R, V_xi_]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_xi_))),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_))),s(TV_u_27b,h4s_bools_arb)))).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
fof(ah4s_lists_listu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27listu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => 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,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ch4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_u_UNIQUEu_c0, conjecture, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_2]: (![V_P, V_x, V_y]: 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)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_fun(TV_u_27a,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))),s(TV_u_27a,V_y))) = 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_x]: 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))) = s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27c,t_fun(TV_u_27a,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_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))))))))))).
