%   ORIGINAL: h4/Omega__Automata/OMEGA__CONJ__CLOSURE
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Goal: !t0 i Psi2 Psi1 Phi__R2 Phi__R1 Phi__I2 Phi__I1. (?q1. Phi__I1 (q1 t0) /\ (!t. Phi__R1 (h4/pair/_2C (i (h4/arithmetic/_2B t t0)) (q1 (h4/arithmetic/_2B t t0)))) /\ Psi1 (h4/pair/_2C i q1)) /\ (?q2. Phi__I2 (q2 t0) /\ (!t. Phi__R2 (h4/pair/_2C (i (h4/arithmetic/_2B t t0)) (q2 (h4/arithmetic/_2B t t0)))) /\ Psi2 (h4/pair/_2C i q2)) <=> (?q1 q2. (Phi__I1 (q1 t0) /\ Phi__I2 (q2 t0)) /\ (!t. Phi__R1 (h4/pair/_2C (i (h4/arithmetic/_2B t t0)) (q1 (h4/arithmetic/_2B t t0))) /\ Phi__R2 (h4/pair/_2C (i (h4/arithmetic/_2B t t0)) (q2 (h4/arithmetic/_2B t t0)))) /\ Psi1 (h4/pair/_2C i q1) /\ Psi2 (h4/pair/_2C i 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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Goal: !t0 i Psi2 Psi1 Phi__R2 Phi__R1 Phi__I2 Phi__I1. (?q1. happ Phi__I1 (happ q1 t0) /\ (!t. happ Phi__R1 (h4/pair/_2C (happ i (h4/arithmetic/_2B t t0)) (happ q1 (h4/arithmetic/_2B t t0)))) /\ happ Psi1 (h4/pair/_2C i q1)) /\ (?q2. happ Phi__I2 (happ q2 t0) /\ (!t. happ Phi__R2 (h4/pair/_2C (happ i (h4/arithmetic/_2B t t0)) (happ q2 (h4/arithmetic/_2B t t0)))) /\ happ Psi2 (h4/pair/_2C i q2)) <=> (?q1 q2. (happ Phi__I1 (happ q1 t0) /\ happ Phi__I2 (happ q2 t0)) /\ (!t. happ Phi__R1 (h4/pair/_2C (happ i (h4/arithmetic/_2B t t0)) (happ q1 (h4/arithmetic/_2B t t0))) /\ happ Phi__R2 (h4/pair/_2C (happ i (h4/arithmetic/_2B t t0)) (happ q2 (h4/arithmetic/_2B t t0)))) /\ happ Psi1 (h4/pair/_2C i q1) /\ happ Psi2 (h4/pair/_2C i q2))
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_Q215967,TV_Q215963]: ![V_f, V_g]: (![V_x]: s(TV_Q215963,happ(s(t_fun(TV_Q215967,TV_Q215963),V_f),s(TV_Q215967,V_x))) = s(TV_Q215963,happ(s(t_fun(TV_Q215967,TV_Q215963),V_g),s(TV_Q215967,V_x))) => s(t_fun(TV_Q215967,TV_Q215963),V_f) = s(t_fun(TV_Q215967,TV_Q215963),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_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,t1)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t1) <=> p(s(t_bool,V_t)))).
fof(ch4s_Omegau_u_Automatas_OMEGAu_u_CONJu_u_CLOSURE, conjecture, ![TV_u_27b1,TV_u_27a,TV_u_27b2]: ![V_t0, V_i, V_Psi2, V_Psi1, V_Phiu_u_R2, V_Phiu_u_R1, V_Phiu_u_I2, V_Phiu_u_I1]: ((?[V_q1]: (p(s(t_bool,happ(s(t_fun(TV_u_27b1,t_bool),V_Phiu_u_I1),s(TV_u_27b1,happ(s(t_fun(t_h4s_nums_num,TV_u_27b1),V_q1),s(t_h4s_nums_num,V_t0)))))) & (![V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b1),t_bool),V_Phiu_u_R1),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b1),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_i),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0))))),s(TV_u_27b1,happ(s(t_fun(t_h4s_nums_num,TV_u_27b1),V_q1),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_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b1)),t_bool),V_Psi1),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b1)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,TV_u_27a),V_i),s(t_fun(t_h4s_nums_num,TV_u_27b1),V_q1)))))))) & ?[V_q2]: (p(s(t_bool,happ(s(t_fun(TV_u_27b2,t_bool),V_Phiu_u_I2),s(TV_u_27b2,happ(s(t_fun(t_h4s_nums_num,TV_u_27b2),V_q2),s(t_h4s_nums_num,V_t0)))))) & (![V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b2),t_bool),V_Phiu_u_R2),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b2),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_i),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0))))),s(TV_u_27b2,happ(s(t_fun(t_h4s_nums_num,TV_u_27b2),V_q2),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_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b2)),t_bool),V_Psi2),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b2)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,TV_u_27a),V_i),s(t_fun(t_h4s_nums_num,TV_u_27b2),V_q2))))))))) <=> ?[V_q1, V_q2]: ((p(s(t_bool,happ(s(t_fun(TV_u_27b1,t_bool),V_Phiu_u_I1),s(TV_u_27b1,happ(s(t_fun(t_h4s_nums_num,TV_u_27b1),V_q1),s(t_h4s_nums_num,V_t0)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27b2,t_bool),V_Phiu_u_I2),s(TV_u_27b2,happ(s(t_fun(t_h4s_nums_num,TV_u_27b2),V_q2),s(t_h4s_nums_num,V_t0))))))) & (![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b1),t_bool),V_Phiu_u_R1),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b1),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_i),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0))))),s(TV_u_27b1,happ(s(t_fun(t_h4s_nums_num,TV_u_27b1),V_q1),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_pairs_prod(TV_u_27a,TV_u_27b2),t_bool),V_Phiu_u_R2),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b2),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_i),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_t0))))),s(TV_u_27b2,happ(s(t_fun(t_h4s_nums_num,TV_u_27b2),V_q2),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_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b1)),t_bool),V_Psi1),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b1)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,TV_u_27a),V_i),s(t_fun(t_h4s_nums_num,TV_u_27b1),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_27b2)),t_bool),V_Psi2),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27b2)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,TV_u_27a),V_i),s(t_fun(t_h4s_nums_num,TV_u_27b2),V_q2))))))))))).
