%   ORIGINAL: h4/path/not__every
% 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/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/path/every__def: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Goal: !p P. ~h4/path/every P p <=> h4/path/exists (h4/combin/o $not P) p
%   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_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_paths_everyu_u_def]: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Goal: !p P. ~h4/path/every P p <=> h4/path/exists (h4/combin/o $not P) 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_Q141956,TV_Q141952]: ![V_f, V_g]: (![V_x]: s(TV_Q141952,happ(s(t_fun(TV_Q141956,TV_Q141952),V_f),s(TV_Q141956,V_x))) = s(TV_Q141952,happ(s(t_fun(TV_Q141956,TV_Q141952),V_g),s(TV_Q141956,V_x))) => s(t_fun(TV_Q141956,TV_Q141952),V_f) = s(t_fun(TV_Q141956,TV_Q141952),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_paths_everyu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (p(s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ~ (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))).
fof(ch4s_paths_notu_u_every, conjecture, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (~ (p(s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) <=> p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
