%   ORIGINAL: h4/Temporal__Logic/WHEN__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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/Temporal__Logic/NEXT0: !P. h4/Temporal__Logic/NEXT P = (\t. P (h4/num/SUC t))
% Assm: h4/Temporal__Logic/WATCH0: !t0 q b. h4/Temporal__Logic/WATCH q b t0 <=> (!t. (q t0 <=> F) /\ (q (h4/num/SUC (h4/arithmetic/_2B t t0)) <=> q (h4/arithmetic/_2B t t0) \/ b (h4/arithmetic/_2B t t0)))
% Assm: h4/Temporal__Logic/WHEN0: !t0 b a. h4/Temporal__Logic/WHEN a b t0 <=> (?q. h4/Temporal__Logic/WATCH q b t0 /\ (!t. q (h4/arithmetic/_2B t t0) \/ (b (h4/arithmetic/_2B t t0) ==> a (h4/arithmetic/_2B t t0))))
% Goal: !b a. h4/Temporal__Logic/NEXT (h4/Temporal__Logic/WHEN a b) = h4/Temporal__Logic/WHEN (h4/Temporal__Logic/NEXT a) (h4/Temporal__Logic/NEXT b)
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_Temporalu_u_Logics_NEXT0]: !P x. happ (h4/Temporal__Logic/NEXT P) x <=> happ P (h4/num/SUC x)
% Assm [h4s_Temporalu_u_Logics_WATCH0]: !t0 q b. h4/Temporal__Logic/WATCH q b t0 <=> (!t. (happ q t0 <=> F) /\ (happ q (h4/num/SUC (h4/arithmetic/_2B t t0)) <=> happ q (h4/arithmetic/_2B t t0) \/ happ b (h4/arithmetic/_2B t t0)))
% Assm [h4s_Temporalu_u_Logics_WHEN0]: !t0 b a. happ (h4/Temporal__Logic/WHEN a b) t0 <=> (?q. h4/Temporal__Logic/WATCH q b t0 /\ (!t. happ q (h4/arithmetic/_2B t t0) \/ (happ b (h4/arithmetic/_2B t t0) ==> happ a (h4/arithmetic/_2B t t0))))
% Goal: !b a. h4/Temporal__Logic/NEXT (h4/Temporal__Logic/WHEN a b) = h4/Temporal__Logic/WHEN (h4/Temporal__Logic/NEXT a) (h4/Temporal__Logic/NEXT b)
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_Q303597,TV_Q303593]: ![V_f, V_g]: (![V_x]: s(TV_Q303593,happ(s(t_fun(TV_Q303597,TV_Q303593),V_f),s(TV_Q303597,V_x))) = s(TV_Q303593,happ(s(t_fun(TV_Q303597,TV_Q303593),V_g),s(TV_Q303597,V_x))) => s(t_fun(TV_Q303597,TV_Q303593),V_f) = s(t_fun(TV_Q303597,TV_Q303593),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_Temporalu_u_Logics_NEXT0, axiom, ![V_P, V_x]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,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(ah4s_Temporalu_u_Logics_WATCH0, axiom, ![V_t0, V_q, V_b]: (p(s(t_bool,h4s_temporalu_u_logics_watch(s(t_fun(t_h4s_nums_num,t_bool),V_q),s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_t0)))) <=> ![V_t]: (s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_q),s(t_h4s_nums_num,V_t0))) = s(t_bool,f) & (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_q),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0)))))))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_q),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0)))))) | p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),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_WHEN0, axiom, ![V_t0, V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_when(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_fun(t_h4s_nums_num,t_bool),V_b))),s(t_h4s_nums_num,V_t0)))) <=> ?[V_q]: (p(s(t_bool,h4s_temporalu_u_logics_watch(s(t_fun(t_h4s_nums_num,t_bool),V_q),s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_t0)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_q),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0)))))) | (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),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_WHENu_u_NEXT, conjecture, ![V_b, V_a]: s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_next(s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_when(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_fun(t_h4s_nums_num,t_bool),V_b))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_when(s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_next(s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_next(s(t_fun(t_h4s_nums_num,t_bool),V_b)))))).
