%   ORIGINAL: h4/llist/every__thm_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/bool/TRUTH: 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_c0: !t. (T <=> t) <=> t
% Assm: h4/llist/exists__thm_c0: !P. h4/llist/exists P h4/llist/LNIL <=> F
% Assm: h4/llist/every__def: !ll P. h4/llist/every P ll <=> ~h4/llist/exists (h4/combin/o $not P) ll
% Goal: !P. h4/llist/every P h4/llist/LNIL <=> T
%   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_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_llists_existsu_u_thmu_c0]: !P. h4/llist/exists P h4/llist/LNIL <=> F
% Assm [h4s_llists_everyu_u_def]: !ll P. h4/llist/every P ll <=> ~h4/llist/exists (h4/combin/o $not P) ll
% Goal: !P. h4/llist/every P h4/llist/LNIL <=> T
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_Q171275,TV_Q171271]: ![V_f, V_g]: (![V_x]: s(TV_Q171271,happ(s(t_fun(TV_Q171275,TV_Q171271),V_f),s(TV_Q171275,V_x))) = s(TV_Q171271,happ(s(t_fun(TV_Q171275,TV_Q171271),V_g),s(TV_Q171275,V_x))) => s(t_fun(TV_Q171275,TV_Q171271),V_f) = s(t_fun(TV_Q171275,TV_Q171271),V_g))).
fof(ah4s_bools_TRUTH, axiom, 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_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_llists_existsu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_llists_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_bool,f)).
fof(ah4s_llists_everyu_u_def, axiom, ![TV_u_27a]: ![V_ll, V_P]: (p(s(t_bool,h4s_llists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_ll)))) <=> ~ (p(s(t_bool,h4s_llists_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_llists_llist(TV_u_27a),V_ll))))))).
fof(ch4s_llists_everyu_u_thmu_c0, conjecture, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_llists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_bool,t)).
