%   ORIGINAL: h4/Temporal__Logic/NOT__ALWAYS
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/Temporal__Logic/ALWAYS0: !t0 P. h4/Temporal__Logic/ALWAYS P t0 <=> (!t. P (h4/arithmetic/_2B t t0))
% Assm: h4/Temporal__Logic/EVENTUAL0: !t0 P. h4/Temporal__Logic/EVENTUAL P t0 <=> (?t. P (h4/arithmetic/_2B t t0))
% Goal: !t0 a. ~h4/Temporal__Logic/ALWAYS a t0 <=> h4/Temporal__Logic/EVENTUAL (\t. ~a t) t0
%   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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_Temporalu_u_Logics_ALWAYS0]: !t0 P. h4/Temporal__Logic/ALWAYS P t0 <=> (!t. happ P (h4/arithmetic/_2B t t0))
% Assm [h4s_Temporalu_u_Logics_EVENTUAL0]: !t0 P. h4/Temporal__Logic/EVENTUAL P t0 <=> (?t. happ P (h4/arithmetic/_2B t t0))
% Goal: !_0. (!a t. happ (happ _0 a) t <=> ~happ a t) ==> (!t0 a. ~h4/Temporal__Logic/ALWAYS a t0 <=> h4/Temporal__Logic/EVENTUAL (happ _0 a) t0)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t1))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t1) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q304497,TV_Q304493]: ![V_f, V_g]: (![V_x]: s(TV_Q304493,happ(s(t_fun(TV_Q304497,TV_Q304493),V_f),s(TV_Q304497,V_x))) = s(TV_Q304493,happ(s(t_fun(TV_Q304497,TV_Q304493),V_g),s(TV_Q304497,V_x))) => s(t_fun(TV_Q304497,TV_Q304493),V_f) = s(t_fun(TV_Q304497,TV_Q304493),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t1))).
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_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,t1)))).
fof(ah4s_Temporalu_u_Logics_ALWAYS0, axiom, ![V_t0, V_P]: (p(s(t_bool,h4s_temporalu_u_logics_always(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_t0)))) <=> ![V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0)))))))).
fof(ah4s_Temporalu_u_Logics_EVENTUAL0, axiom, ![V_t0, V_P]: (p(s(t_bool,h4s_temporalu_u_logics_eventual(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_t0)))) <=> ?[V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0)))))))).
fof(ch4s_Temporalu_u_Logics_NOTu_u_ALWAYS, conjecture, ![V_uu_0]: (![V_a, V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_h4s_nums_num,V_t)))) <=> ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_t)))))) => ![V_t0, V_a]: (~ (p(s(t_bool,h4s_temporalu_u_logics_always(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_t0))))) <=> p(s(t_bool,h4s_temporalu_u_logics_eventual(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_h4s_nums_num,V_t0))))))).
