%   ORIGINAL: h4/quantHeuristics/GUESS__RULES__EXISTS__UNIQUE_c1
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EXISTS__UNIQUE__THM: !P. h4/bool/_3F_21 (\x. P x) <=> (?x. P x) /\ (!x y. P x /\ P y ==> x = y)
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/quantHeuristics/GUESS__REWRITES_c3: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~P (i fv))
% Assm: h4/quantHeuristics/GUESS__REWRITES_c4: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P <=> (!v. P v ==> (?fv. v = i fv))
% Goal: !i P. (!y. h4/quantHeuristics/GUESS__EXISTS__GAP i (\x. P x y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP 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_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> 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_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_THM]: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. h4/bool/_3F_21 (happ _0 P) <=> (?x. happ P x) /\ (!x y. happ P x /\ happ P y ==> x = y))
% Assm [h4s_bools_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_quantHeuristicss_GUESSu_u_REWRITESu_c3]: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~happ P (happ i fv))
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c4]: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i P <=> (!v. happ P v ==> (?fv. v = happ i fv))
% 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__EXISTS__GAP i (happ (happ _0 P) y)) ==> h4/quantHeuristics/GUESS__EXISTS__GAP 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_Q22995,TV_Q22991]: ![V_f, V_g]: (![V_x]: s(TV_Q22991,happ(s(t_fun(TV_Q22995,TV_Q22991),V_f),s(TV_Q22995,V_x))) = s(TV_Q22991,happ(s(t_fun(TV_Q22995,TV_Q22991),V_g),s(TV_Q22995,V_x))) => s(t_fun(TV_Q22995,TV_Q22991),V_f) = s(t_fun(TV_Q22995,TV_Q22991),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_EXISTSu_u_UNIQUEu_u_THM, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))) => ![V_P]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P)))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_bools_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_quantHeuristicss_GUESSu_u_REWRITESu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_fv]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_REWRITESu_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(ch4s_quantHeuristicss_GUESSu_u_RULESu_u_EXISTSu_u_UNIQUEu_c1, 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_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))))))))))).
