%   ORIGINAL: h4/Temporal__Logic/EVENTUAL__AS__UNTIL
% 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/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% 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/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/ABS__SIMP: !t2 t1. (\x. t1) t2 = t1
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% 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_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/prim__rec/num__Axiom__old: !f e. h4/bool/_3F_21 (\fn1. fn1 h4/num/0 = e /\ (!n. fn1 (h4/num/SUC n) = f (fn1 n) n))
% Assm: h4/arithmetic/ADD_c1: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__CLAUSES_c2: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ADD__SUB: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% Assm: h4/arithmetic/SUB__EQUAL__0: !c. h4/arithmetic/_2D c c = h4/num/0
% Assm: h4/Temporal__Logic/ALWAYS0: !t0 P. h4/Temporal__Logic/ALWAYS P t0 <=> (!t. P (h4/arithmetic/_2B t t0))
% Assm: h4/Temporal__Logic/EVENTUAL0: !t0 P. h4/Temporal__Logic/EVENTUAL P t0 <=> (?t. P (h4/arithmetic/_2B t t0))
% 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))))
% Assm: h4/Temporal__Logic/UNTIL0: !t0 b a. h4/Temporal__Logic/UNTIL 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: !a. h4/Temporal__Logic/EVENTUAL a = (\t. ~h4/Temporal__Logic/UNTIL (\t0. ~a t0) (\t0. F) 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_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% 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_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_ABSu_u_SIMP]: !t2 t1. t1 = t1
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% 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_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_primu_u_recs_numu_u_Axiomu_u_old]: !_0. (!e f fn1. happ (happ (happ _0 e) f) fn1 <=> happ fn1 h4/num/0 = e /\ (!n. happ fn1 (h4/num/SUC n) = happ (happ f (happ fn1 n)) n)) ==> (!f e. h4/bool/_3F_21 (happ (happ _0 e) f))
% Assm [h4s_arithmetics_ADDu_c1]: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c2]: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_ADDu_u_SUB]: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% Assm [h4s_arithmetics_SUBu_u_EQUALu_u_0]: !c. h4/arithmetic/_2D c c = h4/num/0
% Assm [h4s_Temporalu_u_Logics_ALWAYS0]: !t0 P. h4/Temporal__Logic/ALWAYS P t0 <=> (!t. happ P (h4/arithmetic/_2B t t0))
% Assm [h4s_Temporalu_u_Logics_EVENTUAL0]: !t0 P. h4/Temporal__Logic/EVENTUAL P t0 <=> (?t. happ P (h4/arithmetic/_2B t t0))
% 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. 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))))
% Assm [h4s_Temporalu_u_Logics_UNTIL0]: !t0 b a. h4/Temporal__Logic/UNTIL 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: !_1. (!t0. happ _1 t0 <=> F) ==> (!_0. (!a t0. happ (happ _0 a) t0 <=> ~happ a t0) ==> (!a x. h4/Temporal__Logic/EVENTUAL a x <=> ~h4/Temporal__Logic/UNTIL (happ _0 a) _1 x))
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_Q301319,TV_Q301315]: ![V_f, V_g]: (![V_x]: s(TV_Q301315,happ(s(t_fun(TV_Q301319,TV_Q301315),V_f),s(TV_Q301319,V_x))) = s(TV_Q301315,happ(s(t_fun(TV_Q301319,TV_Q301315),V_g),s(TV_Q301319,V_x))) => s(t_fun(TV_Q301319,TV_Q301315),V_f) = s(t_fun(TV_Q301319,TV_Q301315),V_g))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_ABSu_u_SIMP, axiom, ![TV_u_27b,TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,V_t1) = s(TV_u_27a,V_t1)).
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_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> 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,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(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_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_primu_u_recs_numu_u_Axiomu_u_old, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_e, V_f, V_fn1]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool))),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn1)))) <=> (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn1),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn1),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn1),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_n))))) => ![V_f, V_e]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool))),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f)))))))).
fof(ah4s_arithmetics_ADDu_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),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_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),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_arithmetics_ADDu_u_SUB, axiom, ![V_c, V_a]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_c))),s(t_h4s_nums_num,V_c))) = s(t_h4s_nums_num,V_a)).
fof(ah4s_arithmetics_SUBu_u_EQUALu_u_0, axiom, ![V_c]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_c),s(t_h4s_nums_num,V_c))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_Temporalu_u_Logics_ALWAYS0, axiom, ![V_t0, V_P]: (p(s(t_bool,h4s_temporalu_u_logics_always(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_t0)))) <=> ![V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),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_EVENTUAL0, axiom, ![V_t0, V_P]: (p(s(t_bool,h4s_temporalu_u_logics_eventual(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_t0)))) <=> ?[V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),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_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,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(ah4s_Temporalu_u_Logics_UNTIL0, axiom, ![V_t0, V_b, V_a]: (p(s(t_bool,h4s_temporalu_u_logics_until(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_EVENTUALu_u_ASu_u_UNTIL, conjecture, ![V_uu_1]: (![V_t0]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_uu_1),s(t_h4s_nums_num,V_t0))) = s(t_bool,f) => ![V_uu_0]: (![V_a, V_t0]: (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)),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_a))),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,V_t0)))))) => ![V_a, V_x]: (p(s(t_bool,h4s_temporalu_u_logics_eventual(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) <=> ~ (p(s(t_bool,h4s_temporalu_u_logics_until(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)),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_fun(t_h4s_nums_num,t_bool),V_uu_1),s(t_h4s_nums_num,V_x))))))))).
