%   ORIGINAL: h4/Temporal__Logic/IMP__NEXT
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/Temporal__Logic/NEXT0: !P. h4/Temporal__Logic/NEXT P = (\t. P (h4/num/SUC t))
% Goal: !Q P. h4/Temporal__Logic/NEXT (\t. P t ==> Q t) = (\t. h4/Temporal__Logic/NEXT P t ==> h4/Temporal__Logic/NEXT Q 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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_Temporalu_u_Logics_NEXT0]: !P x. h4/Temporal__Logic/NEXT P x <=> happ P (h4/num/SUC x)
% Goal: !_0. (!P Q t. happ (happ (happ _0 P) Q) t <=> happ P t ==> happ Q t) ==> (!Q P x. h4/Temporal__Logic/NEXT (happ (happ _0 P) Q) x <=> h4/Temporal__Logic/NEXT P x ==> h4/Temporal__Logic/NEXT Q x)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q303497,TV_Q303493]: ![V_f, V_g]: (![V_x]: s(TV_Q303493,happ(s(t_fun(TV_Q303497,TV_Q303493),V_f),s(TV_Q303497,V_x))) = s(TV_Q303493,happ(s(t_fun(TV_Q303497,TV_Q303493),V_g),s(TV_Q303497,V_x))) => s(t_fun(TV_Q303497,TV_Q303493),V_f) = s(t_fun(TV_Q303497,TV_Q303493),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
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,t0)))).
fof(ah4s_Temporalu_u_Logics_NEXT0, axiom, ![V_P, V_x]: s(t_bool,h4s_temporalu_u_logics_next(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_x))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_x)))))).
fof(ch4s_Temporalu_u_Logics_IMPu_u_NEXT, conjecture, ![V_uu_0]: (![V_P, V_Q, 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)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),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_P))),s(t_fun(t_h4s_nums_num,t_bool),V_Q))),s(t_h4s_nums_num,V_t)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_t)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,V_t)))))) => ![V_Q, V_P, V_x]: (p(s(t_bool,h4s_temporalu_u_logics_next(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)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),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_P))),s(t_fun(t_h4s_nums_num,t_bool),V_Q))),s(t_h4s_nums_num,V_x)))) <=> (p(s(t_bool,h4s_temporalu_u_logics_next(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_x)))) => p(s(t_bool,h4s_temporalu_u_logics_next(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,V_x)))))))).
