%   ORIGINAL: h4/path/every__thm_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/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/path/every__def: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Assm: h4/path/exists__thm_c1: !x r p P. h4/path/exists P (h4/path/pcons x r p) <=> P x \/ h4/path/exists P p
% Goal: !x r p P. h4/path/every P (h4/path/pcons x r p) <=> P x /\ h4/path/every 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_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_paths_everyu_u_def]: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Assm [h4s_paths_existsu_u_thmu_c1]: !x r p P. h4/path/exists P (h4/path/pcons x r p) <=> happ P x \/ h4/path/exists P p
% Goal: !x r p P. h4/path/every P (h4/path/pcons x r p) <=> happ P x /\ h4/path/every 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_Q141931,TV_Q141927]: ![V_f, V_g]: (![V_x]: s(TV_Q141927,happ(s(t_fun(TV_Q141931,TV_Q141927),V_f),s(TV_Q141931,V_x))) = s(TV_Q141927,happ(s(t_fun(TV_Q141931,TV_Q141927),V_g),s(TV_Q141931,V_x))) => s(t_fun(TV_Q141931,TV_Q141927),V_f) = s(t_fun(TV_Q141931,TV_Q141927),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_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_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_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(ah4s_paths_existsu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27c]: ![V_x, V_r, V_p, V_P]: (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27c,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),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,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),V_p))))))).
fof(ch4s_paths_everyu_u_thmu_c1, conjecture, ![TV_u_27a,TV_u_27c]: ![V_x, V_r, 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_27c),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27c,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),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,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),V_p))))))).
