%   ORIGINAL: h4/quantHeuristics/GUESSES__NEG__REWRITE_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/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c1: !i P. h4/quantHeuristics/GUESS__FORALL i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c2: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__GAP i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c3: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS__GAP i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c4: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__POINT i P
% Assm: h4/quantHeuristics/GUESSES__NEG__DUALITY_c5: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS__POINT i P
% Goal: !i P. h4/quantHeuristics/GUESS__EXISTS i (\x. ~P x) <=> h4/quantHeuristics/GUESS__FORALL i (\x. P x)
%   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_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c0]: !i P. h4/quantHeuristics/GUESS__EXISTS i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c1]: !i P. h4/quantHeuristics/GUESS__FORALL i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c2]: !i P. h4/quantHeuristics/GUESS__EXISTS__GAP i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__GAP i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c3]: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS__GAP i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c4]: !i P. h4/quantHeuristics/GUESS__EXISTS__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__FORALL__POINT i P
% Assm [h4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c5]: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i (h4/combin/o $not P) <=> h4/quantHeuristics/GUESS__EXISTS__POINT i P
% Goal: !_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)))
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_Q20220,TV_Q20216]: ![V_f, V_g]: (![V_x]: s(TV_Q20216,happ(s(t_fun(TV_Q20220,TV_Q20216),V_f),s(TV_Q20220,V_x))) = s(TV_Q20216,happ(s(t_fun(TV_Q20220,TV_Q20216),V_g),s(TV_Q20220,V_x))) => s(t_fun(TV_Q20220,TV_Q20216),V_f) = s(t_fun(TV_Q20220,TV_Q20216),V_g))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c4, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ah4s_quantHeuristicss_GUESSESu_u_NEGu_u_DUALITYu_c5, axiom, ![TV_u_27a,TV_u_27b]: ![V_i, V_P]: s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_bool,h4s_quantheuristicss_guessu_u_existsu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))).
fof(ch4s_quantHeuristicss_GUESSESu_u_NEGu_u_REWRITEu_c0, conjecture, ![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)))))))).
