%   ORIGINAL: h4/Omega__Automata/AUTOMATON__TEMP__CLOSURE_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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Goal: !Phi__R2 Phi__R1 Phi__I2 Phi__I1 Phi__F. (?q1. Phi__I1 q1 /\ Phi__R1 q1 /\ (?q2. Phi__I2 q2 /\ Phi__R2 (h4/pair/_2C q2 q1) /\ Phi__F (h4/pair/_2C q1 q2))) <=> (?q1 q2. (Phi__I1 q1 /\ Phi__I2 q2) /\ (Phi__R1 q1 /\ Phi__R2 (h4/pair/_2C q2 q1)) /\ Phi__F (h4/pair/_2C q1 q2))
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Goal: !Phi__R2 Phi__R1 Phi__I2 Phi__I1 Phi__F. (?q1. happ Phi__I1 q1 /\ happ Phi__R1 q1 /\ (?q2. happ Phi__I2 q2 /\ happ Phi__R2 (h4/pair/_2C q2 q1) /\ happ Phi__F (h4/pair/_2C q1 q2))) <=> (?q1 q2. (happ Phi__I1 q1 /\ happ Phi__I2 q2) /\ (happ Phi__R1 q1 /\ happ Phi__R2 (h4/pair/_2C q2 q1)) /\ happ Phi__F (h4/pair/_2C q1 q2))
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_Q216969,TV_Q216965]: ![V_f, V_g]: (![V_x]: s(TV_Q216965,happ(s(t_fun(TV_Q216969,TV_Q216965),V_f),s(TV_Q216969,V_x))) = s(TV_Q216965,happ(s(t_fun(TV_Q216969,TV_Q216965),V_g),s(TV_Q216969,V_x))) => s(t_fun(TV_Q216969,TV_Q216965),V_f) = s(t_fun(TV_Q216969,TV_Q216965),V_g))).
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_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_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(ch4s_Omegau_u_Automatas_AUTOMATONu_u_TEMPu_u_CLOSUREu_c0, conjecture, ![TV_u_27a,TV_u_27b]: ![V_Phiu_u_R2, V_Phiu_u_R1, V_Phiu_u_I2, V_Phiu_u_I1, V_Phiu_u_F]: (?[V_q1]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool),V_Phiu_u_I1),s(t_fun(t_h4s_nums_num,TV_u_27a),V_q1)))) & (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool),V_Phiu_u_R1),s(t_fun(t_h4s_nums_num,TV_u_27a),V_q1)))) & ?[V_q2]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),t_bool),V_Phiu_u_I2),s(t_fun(t_h4s_nums_num,TV_u_27b),V_q2)))) & (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a)),t_bool),V_Phiu_u_R2),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,TV_u_27b),V_q2),s(t_fun(t_h4s_nums_num,TV_u_27a),V_q1)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b)),t_bool),V_Phiu_u_F),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,TV_u_27a),V_q1),s(t_fun(t_h4s_nums_num,TV_u_27b),V_q2)))))))))) <=> ?[V_q1, V_q2]: ((p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool),V_Phiu_u_I1),s(t_fun(t_h4s_nums_num,TV_u_27a),V_q1)))) & p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),t_bool),V_Phiu_u_I2),s(t_fun(t_h4s_nums_num,TV_u_27b),V_q2))))) & ((p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool),V_Phiu_u_R1),s(t_fun(t_h4s_nums_num,TV_u_27a),V_q1)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a)),t_bool),V_Phiu_u_R2),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,TV_u_27b),V_q2),s(t_fun(t_h4s_nums_num,TV_u_27a),V_q1))))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b)),t_bool),V_Phiu_u_F),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,TV_u_27a),V_q1),s(t_fun(t_h4s_nums_num,TV_u_27b),V_q2)))))))))).
