%   ORIGINAL: h4/Omega__Automata/CO__BUECHI__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/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/LESS__CASES: !n m. h4/prim__rec/_3C m n \/ h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/LESS__ADD__1: !n m. h4/prim__rec/_3C n m ==> (?p. m = h4/arithmetic/_2B n (h4/arithmetic/_2B p (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/arithmetic/LESS__EQUAL__ADD: !n m. h4/arithmetic/_3C_3D m n ==> (?p. n = h4/arithmetic/_2B m p)
% 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)))) /\ (?t1. !t2. Psi1 (h4/pair/_2C (i (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))) (q1 (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0)))))) /\ (?q2. Phi__I2 (q2 t0) /\ (!t. Phi__R2 (h4/pair/_2C (i (h4/arithmetic/_2B t t0)) (q2 (h4/arithmetic/_2B t t0)))) /\ (?t1. !t2. Psi2 (h4/pair/_2C (i (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))) (q2 (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0)))))) <=> (?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)))) /\ (?t1. !t2. Psi1 (h4/pair/_2C (i (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))) (q1 (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0)))) /\ Psi2 (h4/pair/_2C (i (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))) (q2 (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 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_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~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_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_LESSu_u_CASES]: !n m. h4/prim__rec/_3C m n \/ h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_LESSu_u_ADDu_u_1]: !n m. h4/prim__rec/_3C n m ==> (?p. m = h4/arithmetic/_2B n (h4/arithmetic/_2B p (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_arithmetics_LESSu_u_EQUALu_u_ADD]: !n m. h4/arithmetic/_3C_3D m n ==> (?p. n = h4/arithmetic/_2B m p)
% 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)))) /\ (?t1. !t2. happ Psi1 (h4/pair/_2C (happ i (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))) (happ q1 (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0)))))) /\ (?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)))) /\ (?t1. !t2. happ Psi2 (h4/pair/_2C (happ i (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))) (happ q2 (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0)))))) <=> (?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)))) /\ (?t1. !t2. happ Psi1 (h4/pair/_2C (happ i (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))) (happ q1 (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0)))) /\ happ Psi2 (h4/pair/_2C (happ i (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))) (happ q2 (h4/arithmetic/_2B t1 (h4/arithmetic/_2B t2 t0))))))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t3))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t3) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q217494,TV_Q217490]: ![V_f, V_g]: (![V_x]: s(TV_Q217490,happ(s(t_fun(TV_Q217494,TV_Q217490),V_f),s(TV_Q217494,V_x))) = s(TV_Q217490,happ(s(t_fun(TV_Q217494,TV_Q217490),V_g),s(TV_Q217494,V_x))) => s(t_fun(TV_Q217494,TV_Q217490),V_f) = s(t_fun(TV_Q217494,TV_Q217490),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t3))).
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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (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,t3)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t3))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t3)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t3) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, 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_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_CASES, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_LESSu_u_ADDu_u_1, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) => ?[V_p]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))).
fof(ah4s_arithmetics_LESSu_u_EQUALu_u_ADD, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ?[V_p]: s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))).
fof(ch4s_Omegau_u_Automatas_COu_u_BUECHIu_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)))))))))) & ?[V_t1]: ![V_t2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b1),t_bool),V_Psi1),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_t1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t2),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_t1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t2),s(t_h4s_nums_num,V_t0)))))))))))))) & ?[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)))))))))) & ?[V_t1]: ![V_t2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b2),t_bool),V_Psi2),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_t1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t2),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_t1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t2),s(t_h4s_nums_num,V_t0))))))))))))))) <=> ?[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))))))))))) & ?[V_t1]: ![V_t2]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b1),t_bool),V_Psi1),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_t1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t2),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_t1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t2),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_Psi2),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_t1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t2),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_t1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t2),s(t_h4s_nums_num,V_t0))))))))))))))))).
