%   ORIGINAL: h4/Omega__Automata/TEMP__OPS__DEFS__TO__OMEGA_c5
% 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/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/EXISTS__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/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/OR__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_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> 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__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> 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/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~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/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/prim__rec/LESS__REFL: !n. ~h4/prim__rec/_3C n n
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/prim__rec/PRE0_c0: h4/prim__rec/PRE h4/num/0 = h4/num/0
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = 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/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% 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__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ADD1: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% 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/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/EQ__LESS__EQ: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/Temporal__Logic/NEXT0: !P. h4/Temporal__Logic/NEXT P = (\t. P (h4/num/SUC t))
% Assm: h4/Temporal__Logic/EVENTUAL0: !t0 P. h4/Temporal__Logic/EVENTUAL P t0 <=> (?t. P (h4/arithmetic/_2B t t0))
% Assm: h4/Temporal__Logic/DELTA__CASES: !t0 b. (?d. (!t. h4/prim__rec/_3C t d ==> ~b (h4/arithmetic/_2B t t0)) /\ b (h4/arithmetic/_2B d t0)) \/ (!d. ~b (h4/arithmetic/_2B d t0))
% Assm: h4/Temporal__Logic/WHEN__SIGNAL: !t0 b a. h4/Temporal__Logic/WHEN a b t0 <=> (!delta. (!t. h4/prim__rec/_3C t delta ==> ~b (h4/arithmetic/_2B t t0)) /\ b (h4/arithmetic/_2B delta t0) ==> a (h4/arithmetic/_2B delta t0))
% Assm: h4/Temporal__Logic/SWHEN__SIGNAL: !t0 b a. h4/Temporal__Logic/SWHEN a b t0 <=> (?delta. (!t. h4/prim__rec/_3C t delta ==> ~b (h4/arithmetic/_2B t t0)) /\ b (h4/arithmetic/_2B delta t0) /\ a (h4/arithmetic/_2B delta t0))
% Assm: h4/Temporal__Logic/UNTIL__AS__WHEN: !b a. h4/Temporal__Logic/UNTIL a b = h4/Temporal__Logic/WHEN b (\t. a t ==> b t)
% Assm: h4/Temporal__Logic/ALWAYS__AS__UNTIL: !a. h4/Temporal__Logic/ALWAYS a = h4/Temporal__Logic/UNTIL a (\t. F)
% Assm: h4/Temporal__Logic/SBEFORE__AS__UNTIL: !b a. h4/Temporal__Logic/SBEFORE a b = (\t0. ~h4/Temporal__Logic/UNTIL (\t. ~a t) b t0)
% Assm: h4/Temporal__Logic/SUNTIL__AS__SWHEN: !b a. h4/Temporal__Logic/SUNTIL a b = h4/Temporal__Logic/SWHEN b (\t. a t ==> b t)
% Assm: h4/Temporal__Logic/EVENTUAL__AS__SUNTIL: !a. h4/Temporal__Logic/EVENTUAL a = h4/Temporal__Logic/SUNTIL (\t. T) a
% Assm: h4/Temporal__Logic/BEFORE__AS__SUNTIL: !b a. h4/Temporal__Logic/BEFORE a b = (\t. ~h4/Temporal__Logic/SUNTIL (\t0. ~a t0) b t)
% Assm: h4/Temporal__Logic/SOME__EVENT_c0: !t0 b. h4/Temporal__Logic/EVENTUAL b t0 <=> (!a. h4/Temporal__Logic/WHEN a b t0 <=> h4/Temporal__Logic/SWHEN a b t0)
% Assm: h4/Temporal__Logic/WHEN__FIX: !y b a. y = (\t. h4/bool/COND (b t) (a t) (y (h4/arithmetic/_2B t (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) <=> y = h4/Temporal__Logic/WHEN a b \/ y = h4/Temporal__Logic/SWHEN a b
% Assm: h4/Past__Temporal__Logic/PSNEXT0: !t0 a. h4/Past__Temporal__Logic/PSNEXT a t0 <=> h4/prim__rec/_3C h4/num/0 t0 /\ a (h4/prim__rec/PRE t0)
% Assm: h4/Past__Temporal__Logic/PNEXT0: !t0 a. h4/Past__Temporal__Logic/PNEXT a t0 <=> t0 = h4/num/0 \/ a (h4/prim__rec/PRE t0)
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c0: !a. h4/Past__Temporal__Logic/PNEXT a h4/num/0 <=> T
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c1: !a. h4/Past__Temporal__Logic/PSNEXT a h4/num/0 <=> F
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c2: !a. h4/Past__Temporal__Logic/PALWAYS a h4/num/0 <=> a h4/num/0
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c3: !a. h4/Past__Temporal__Logic/PEVENTUAL a h4/num/0 <=> a h4/num/0
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c4: !b a. h4/Past__Temporal__Logic/PSUNTIL a b h4/num/0 <=> b h4/num/0
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c5: !b a. h4/Past__Temporal__Logic/PSWHEN a b h4/num/0 <=> a h4/num/0 /\ b h4/num/0
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c6: !b a. h4/Past__Temporal__Logic/PSBEFORE a b h4/num/0 <=> a h4/num/0 /\ ~b h4/num/0
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c7: !b a. h4/Past__Temporal__Logic/PUNTIL a b h4/num/0 <=> a h4/num/0 \/ b h4/num/0
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c8: !b a. h4/Past__Temporal__Logic/PWHEN a b h4/num/0 <=> a h4/num/0 \/ ~b h4/num/0
% Assm: h4/Past__Temporal__Logic/INITIALISATION_c9: !b a. h4/Past__Temporal__Logic/PBEFORE a b h4/num/0 <=> ~b h4/num/0
% Assm: h4/Past__Temporal__Logic/RECURSION_c8: !a. h4/Past__Temporal__Logic/PALWAYS a = (\t. a t /\ h4/Past__Temporal__Logic/PNEXT (h4/Past__Temporal__Logic/PALWAYS a) t)
% Assm: h4/Past__Temporal__Logic/RECURSION_c9: !a. h4/Past__Temporal__Logic/PEVENTUAL a = (\t. a t \/ h4/Past__Temporal__Logic/PSNEXT (h4/Past__Temporal__Logic/PEVENTUAL a) t)
% Assm: h4/Past__Temporal__Logic/RECURSION_c10: !b a. h4/Past__Temporal__Logic/PSUNTIL a b = (\t. b t \/ a t /\ h4/Past__Temporal__Logic/PSNEXT (h4/Past__Temporal__Logic/PSUNTIL a b) t)
% Assm: h4/Past__Temporal__Logic/RECURSION_c11: !b a. h4/Past__Temporal__Logic/PSWHEN a b = (\t. a t /\ b t \/ ~b t /\ h4/Past__Temporal__Logic/PSNEXT (h4/Past__Temporal__Logic/PSWHEN a b) t)
% Assm: h4/Past__Temporal__Logic/RECURSION_c12: !b a. h4/Past__Temporal__Logic/PSBEFORE a b = (\t. ~b t /\ (a t \/ h4/Past__Temporal__Logic/PSNEXT (h4/Past__Temporal__Logic/PSBEFORE a b) t))
% Assm: h4/Past__Temporal__Logic/RECURSION_c13: !b a. h4/Past__Temporal__Logic/PUNTIL a b = (\t. b t \/ a t /\ h4/Past__Temporal__Logic/PNEXT (h4/Past__Temporal__Logic/PUNTIL a b) t)
% Assm: h4/Past__Temporal__Logic/RECURSION_c14: !b a. h4/Past__Temporal__Logic/PWHEN a b = (\t. a t /\ b t \/ ~b t /\ h4/Past__Temporal__Logic/PNEXT (h4/Past__Temporal__Logic/PWHEN a b) t)
% Assm: h4/Past__Temporal__Logic/RECURSION_c15: !b a. h4/Past__Temporal__Logic/PBEFORE a b = (\t. ~b t /\ (a t \/ h4/Past__Temporal__Logic/PNEXT (h4/Past__Temporal__Logic/PBEFORE a b) t))
% Goal: !l b a. l = h4/Temporal__Logic/SBEFORE a b <=> T /\ (!t. l t <=> ~b t /\ (a t \/ l (h4/num/SUC t))) /\ (!t1. ?t2. l (h4/arithmetic/_2B t1 t2) ==> a (h4/arithmetic/_2B t1 t2) \/ b (h4/arithmetic/_2B t1 t2))
%   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_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_EXISTSu_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_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_ORu_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_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> 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_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> 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_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~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_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% 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_LESSu_u_REFL]: !n. ~h4/prim__rec/_3C n n
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_primu_u_recs_PRE0u_c0]: h4/prim__rec/PRE h4/num/0 = h4/num/0
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = 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_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% 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_EQ]: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ADD1]: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% 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_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_EQu_u_LESSu_u_EQ]: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_Temporalu_u_Logics_NEXT0]: !P x. h4/Temporal__Logic/NEXT P x <=> happ P (h4/num/SUC x)
% Assm [h4s_Temporalu_u_Logics_EVENTUAL0]: !t0 P. happ (h4/Temporal__Logic/EVENTUAL P) t0 <=> (?t. happ P (h4/arithmetic/_2B t t0))
% Assm [h4s_Temporalu_u_Logics_DELTAu_u_CASES]: !t0 b. (?d. (!t. h4/prim__rec/_3C t d ==> ~happ b (h4/arithmetic/_2B t t0)) /\ happ b (h4/arithmetic/_2B d t0)) \/ (!d. ~happ b (h4/arithmetic/_2B d t0))
% Assm [h4s_Temporalu_u_Logics_WHENu_u_SIGNAL]: !t0 b a. happ (h4/Temporal__Logic/WHEN a b) t0 <=> (!delta. (!t. h4/prim__rec/_3C t delta ==> ~happ b (h4/arithmetic/_2B t t0)) /\ happ b (h4/arithmetic/_2B delta t0) ==> happ a (h4/arithmetic/_2B delta t0))
% Assm [h4s_Temporalu_u_Logics_SWHENu_u_SIGNAL]: !t0 b a. happ (h4/Temporal__Logic/SWHEN a b) t0 <=> (?delta. (!t. h4/prim__rec/_3C t delta ==> ~happ b (h4/arithmetic/_2B t t0)) /\ happ b (h4/arithmetic/_2B delta t0) /\ happ a (h4/arithmetic/_2B delta t0))
% Assm [h4s_Temporalu_u_Logics_UNTILu_u_ASu_u_WHEN]: !_0. (!a b t. happ (happ (happ _0 a) b) t <=> happ a t ==> happ b t) ==> (!b a. h4/Temporal__Logic/UNTIL a b = h4/Temporal__Logic/WHEN b (happ (happ _0 a) b))
% Assm [h4s_Temporalu_u_Logics_ALWAYSu_u_ASu_u_UNTIL]: !_0. (!t. happ _0 t <=> F) ==> (!a. h4/Temporal__Logic/ALWAYS a = h4/Temporal__Logic/UNTIL a _0)
% Assm [h4s_Temporalu_u_Logics_SBEFOREu_u_ASu_u_UNTIL]: !_0. (!a t. happ (happ _0 a) t <=> ~happ a t) ==> (!b a x. happ (h4/Temporal__Logic/SBEFORE a b) x <=> ~happ (h4/Temporal__Logic/UNTIL (happ _0 a) b) x)
% Assm [h4s_Temporalu_u_Logics_SUNTILu_u_ASu_u_SWHEN]: !_0. (!a b t. happ (happ (happ _0 a) b) t <=> happ a t ==> happ b t) ==> (!b a. h4/Temporal__Logic/SUNTIL a b = h4/Temporal__Logic/SWHEN b (happ (happ _0 a) b))
% Assm [h4s_Temporalu_u_Logics_EVENTUALu_u_ASu_u_SUNTIL]: !_0. (!t. happ _0 t <=> T) ==> (!a. h4/Temporal__Logic/EVENTUAL a = h4/Temporal__Logic/SUNTIL _0 a)
% Assm [h4s_Temporalu_u_Logics_BEFOREu_u_ASu_u_SUNTIL]: !_0. (!a t0. happ (happ _0 a) t0 <=> ~happ a t0) ==> (!b a x. h4/Temporal__Logic/BEFORE a b x <=> ~happ (h4/Temporal__Logic/SUNTIL (happ _0 a) b) x)
% Assm [h4s_Temporalu_u_Logics_SOMEu_u_EVENTu_c0]: !t0 b. happ (h4/Temporal__Logic/EVENTUAL b) t0 <=> (!a. happ (h4/Temporal__Logic/WHEN a b) t0 <=> happ (h4/Temporal__Logic/SWHEN a b) t0)
% Assm [h4s_Temporalu_u_Logics_WHENu_u_FIX]: !y b a. (!x. happ y x <=> h4/bool/COND (happ b x) (happ a x) (happ y (h4/arithmetic/_2B x (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) <=> y = h4/Temporal__Logic/WHEN a b \/ y = h4/Temporal__Logic/SWHEN a b
% Assm [h4s_Pastu_u_Temporalu_u_Logics_PSNEXT0]: !t0 a. h4/Past__Temporal__Logic/PSNEXT a t0 <=> h4/prim__rec/_3C h4/num/0 t0 /\ happ a (h4/prim__rec/PRE t0)
% Assm [h4s_Pastu_u_Temporalu_u_Logics_PNEXT0]: !t0 a. h4/Past__Temporal__Logic/PNEXT a t0 <=> t0 = h4/num/0 \/ happ a (h4/prim__rec/PRE t0)
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c0]: !a. h4/Past__Temporal__Logic/PNEXT a h4/num/0 <=> T
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c1]: !a. h4/Past__Temporal__Logic/PSNEXT a h4/num/0 <=> F
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c2]: !a. happ (h4/Past__Temporal__Logic/PALWAYS a) h4/num/0 <=> happ a h4/num/0
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c3]: !a. happ (h4/Past__Temporal__Logic/PEVENTUAL a) h4/num/0 <=> happ a h4/num/0
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c4]: !b a. happ (h4/Past__Temporal__Logic/PSUNTIL a b) h4/num/0 <=> happ b h4/num/0
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c5]: !b a. happ (h4/Past__Temporal__Logic/PSWHEN a b) h4/num/0 <=> happ a h4/num/0 /\ happ b h4/num/0
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c6]: !b a. happ (h4/Past__Temporal__Logic/PSBEFORE a b) h4/num/0 <=> happ a h4/num/0 /\ ~happ b h4/num/0
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c7]: !b a. happ (h4/Past__Temporal__Logic/PUNTIL a b) h4/num/0 <=> happ a h4/num/0 \/ happ b h4/num/0
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c8]: !b a. happ (h4/Past__Temporal__Logic/PWHEN a b) h4/num/0 <=> happ a h4/num/0 \/ ~happ b h4/num/0
% Assm [h4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c9]: !b a. happ (h4/Past__Temporal__Logic/PBEFORE a b) h4/num/0 <=> ~happ b h4/num/0
% Assm [h4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c8]: !a x. happ (h4/Past__Temporal__Logic/PALWAYS a) x <=> happ a x /\ h4/Past__Temporal__Logic/PNEXT (h4/Past__Temporal__Logic/PALWAYS a) x
% Assm [h4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c9]: !a x. happ (h4/Past__Temporal__Logic/PEVENTUAL a) x <=> happ a x \/ h4/Past__Temporal__Logic/PSNEXT (h4/Past__Temporal__Logic/PEVENTUAL a) x
% Assm [h4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c10]: !b a x. happ (h4/Past__Temporal__Logic/PSUNTIL a b) x <=> happ b x \/ happ a x /\ h4/Past__Temporal__Logic/PSNEXT (h4/Past__Temporal__Logic/PSUNTIL a b) x
% Assm [h4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c11]: !b a x. happ (h4/Past__Temporal__Logic/PSWHEN a b) x <=> happ a x /\ happ b x \/ ~happ b x /\ h4/Past__Temporal__Logic/PSNEXT (h4/Past__Temporal__Logic/PSWHEN a b) x
% Assm [h4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c12]: !b a x. happ (h4/Past__Temporal__Logic/PSBEFORE a b) x <=> ~happ b x /\ (happ a x \/ h4/Past__Temporal__Logic/PSNEXT (h4/Past__Temporal__Logic/PSBEFORE a b) x)
% Assm [h4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c13]: !b a x. happ (h4/Past__Temporal__Logic/PUNTIL a b) x <=> happ b x \/ happ a x /\ h4/Past__Temporal__Logic/PNEXT (h4/Past__Temporal__Logic/PUNTIL a b) x
% Assm [h4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c14]: !b a x. happ (h4/Past__Temporal__Logic/PWHEN a b) x <=> happ a x /\ happ b x \/ ~happ b x /\ h4/Past__Temporal__Logic/PNEXT (h4/Past__Temporal__Logic/PWHEN a b) x
% Assm [h4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c15]: !b a x. happ (h4/Past__Temporal__Logic/PBEFORE a b) x <=> ~happ b x /\ (happ a x \/ h4/Past__Temporal__Logic/PNEXT (h4/Past__Temporal__Logic/PBEFORE a b) x)
% Goal: !l b a. l = h4/Temporal__Logic/SBEFORE a b <=> T /\ (!t. happ l t <=> ~happ b t /\ (happ a t \/ happ l (h4/num/SUC t))) /\ (!t1. ?t2. happ l (h4/arithmetic/_2B t1 t2) ==> happ a (h4/arithmetic/_2B t1 t2) \/ happ b (h4/arithmetic/_2B t1 t2))
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_Q216619,TV_Q216615]: ![V_f, V_g]: (![V_x]: s(TV_Q216615,happ(s(t_fun(TV_Q216619,TV_Q216615),V_f),s(TV_Q216619,V_x))) = s(TV_Q216615,happ(s(t_fun(TV_Q216619,TV_Q216615),V_g),s(TV_Q216619,V_x))) => s(t_fun(TV_Q216619,TV_Q216615),V_f) = s(t_fun(TV_Q216619,TV_Q216615),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_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_EXISTSu_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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,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_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_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_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => 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_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = 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,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_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
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_conj, 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_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (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_dcu_u_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_s))))) & ((~ (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_s)) | ~ (p(s(t_bool,V_p))))))))))).
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_LESSu_u_REFL, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_PRE0u_c0, axiom, s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
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_ONE, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))).
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_c1, axiom, ![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_0))) = 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_arithmetics_ADDu_u_SYM, 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,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
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_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_NOTu_u_LESS, 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_ADD1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,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_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, 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)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
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_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_EQu_u_LESSu_u_EQ, axiom, ![V_n, V_m]: (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_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_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (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,h4s_nums_suc(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,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_Temporalu_u_Logics_NEXT0, axiom, ![V_P, V_x]: s(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_EVENTUAL0, axiom, ![V_t0, V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,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_DELTAu_u_CASES, axiom, ![V_t0, V_b]: (?[V_d]: (![V_t]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_d)))) => ~ (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_b),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_d),s(t_h4s_nums_num,V_t0))))))) | ![V_d]: ~ (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_d),s(t_h4s_nums_num,V_t0))))))))).
fof(ah4s_Temporalu_u_Logics_WHENu_u_SIGNAL, 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_delta]: ((![V_t]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_delta)))) => ~ (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_b),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_delta),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_delta),s(t_h4s_nums_num,V_t0))))))))).
fof(ah4s_Temporalu_u_Logics_SWHENu_u_SIGNAL, 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_swhen(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_delta]: (![V_t]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_t),s(t_h4s_nums_num,V_delta)))) => ~ (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_b),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_delta),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_delta),s(t_h4s_nums_num,V_t0)))))))))).
fof(ah4s_Temporalu_u_Logics_UNTILu_u_ASu_u_WHEN, axiom, ![V_uu_0]: (![V_a, V_b, V_t]: (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)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),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_b))),s(t_h4s_nums_num,V_t)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_t)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_t)))))) => ![V_b, V_a]: s(t_fun(t_h4s_nums_num,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_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_when(s(t_fun(t_h4s_nums_num,t_bool),V_b),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)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),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_b))))))).
fof(ah4s_Temporalu_u_Logics_ALWAYSu_u_ASu_u_UNTIL, axiom, ![V_uu_0]: (![V_t]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_uu_0),s(t_h4s_nums_num,V_t))) = s(t_bool,f) => ![V_a]: s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_always(s(t_fun(t_h4s_nums_num,t_bool),V_a))) = s(t_fun(t_h4s_nums_num,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_uu_0))))).
fof(ah4s_Temporalu_u_Logics_SBEFOREu_u_ASu_u_UNTIL, axiom, ![V_uu_0]: (![V_a, V_t]: (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_t)))) <=> ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_t)))))) => ![V_b, V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_sbefore(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_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,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_b))),s(t_h4s_nums_num,V_x)))))))).
fof(ah4s_Temporalu_u_Logics_SUNTILu_u_ASu_u_SWHEN, axiom, ![V_uu_0]: (![V_a, V_b, V_t]: (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)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),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_b))),s(t_h4s_nums_num,V_t)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_t)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_t)))))) => ![V_b, V_a]: s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_suntil(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_swhen(s(t_fun(t_h4s_nums_num,t_bool),V_b),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)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),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_b))))))).
fof(ah4s_Temporalu_u_Logics_EVENTUALu_u_ASu_u_SUNTIL, axiom, ![V_uu_0]: (![V_t]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_uu_0),s(t_h4s_nums_num,V_t))) = s(t_bool,t) => ![V_a]: s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_eventual(s(t_fun(t_h4s_nums_num,t_bool),V_a))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_suntil(s(t_fun(t_h4s_nums_num,t_bool),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_a))))).
fof(ah4s_Temporalu_u_Logics_BEFOREu_u_ASu_u_SUNTIL, axiom, ![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_b, V_a, V_x]: (p(s(t_bool,h4s_temporalu_u_logics_before(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_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_suntil(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_b))),s(t_h4s_nums_num,V_x)))))))).
fof(ah4s_Temporalu_u_Logics_SOMEu_u_EVENTu_c0, axiom, ![V_t0, V_b]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_eventual(s(t_fun(t_h4s_nums_num,t_bool),V_b))),s(t_h4s_nums_num,V_t0)))) <=> ![V_a]: 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))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_swhen(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))))).
fof(ah4s_Temporalu_u_Logics_WHENu_u_FIX, axiom, ![V_y, V_b, V_a]: (![V_x]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_y),s(t_h4s_nums_num,V_x))) = s(t_bool,h4s_bools_cond(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x))),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x))),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_y),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))) <=> (s(t_fun(t_h4s_nums_num,t_bool),V_y) = 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),V_y) = s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_swhen(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_fun(t_h4s_nums_num,t_bool),V_b)))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_PSNEXT0, axiom, ![V_t0, V_a]: (p(s(t_bool,h4s_pastu_u_temporalu_u_logics_psnext(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_t0)))) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),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_primu_u_recs_pre(s(t_h4s_nums_num,V_t0))))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_PNEXT0, axiom, ![V_t0, V_a]: (p(s(t_bool,h4s_pastu_u_temporalu_u_logics_pnext(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_t0)))) <=> (s(t_h4s_nums_num,V_t0) = s(t_h4s_nums_num,h4s_nums_0) | p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_t0))))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c0, axiom, ![V_a]: s(t_bool,h4s_pastu_u_temporalu_u_logics_pnext(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,t)).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c1, axiom, ![V_a]: s(t_bool,h4s_pastu_u_temporalu_u_logics_psnext(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,f)).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c2, axiom, ![V_a]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_palways(s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c3, axiom, ![V_a]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_peventual(s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c4, axiom, ![V_b, V_a]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_psuntil(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,h4s_nums_0))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c5, axiom, ![V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pswhen(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,h4s_nums_0)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_nums_0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c6, axiom, ![V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_psbefore(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,h4s_nums_0)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_nums_0)))) & ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c7, axiom, ![V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_puntil(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,h4s_nums_0)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_nums_0)))) | p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c8, axiom, ![V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pwhen(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,h4s_nums_0)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,h4s_nums_0)))) | ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_INITIALISATIONu_c9, axiom, ![V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pbefore(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,h4s_nums_0)))) <=> ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c8, axiom, ![V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_palways(s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_h4s_nums_num,V_x)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) & p(s(t_bool,h4s_pastu_u_temporalu_u_logics_pnext(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_palways(s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_h4s_nums_num,V_x))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c9, axiom, ![V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_peventual(s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_h4s_nums_num,V_x)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) | p(s(t_bool,h4s_pastu_u_temporalu_u_logics_psnext(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_peventual(s(t_fun(t_h4s_nums_num,t_bool),V_a))),s(t_h4s_nums_num,V_x))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c10, axiom, ![V_b, V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_psuntil(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_x)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x)))) | (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) & p(s(t_bool,h4s_pastu_u_temporalu_u_logics_psnext(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_psuntil(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_x)))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c11, axiom, ![V_b, V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pswhen(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_x)))) <=> ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x))))) | (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x))))) & p(s(t_bool,h4s_pastu_u_temporalu_u_logics_psnext(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pswhen(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_x)))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c12, axiom, ![V_b, V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_psbefore(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_x)))) <=> (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) | p(s(t_bool,h4s_pastu_u_temporalu_u_logics_psnext(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_psbefore(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_x)))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c13, axiom, ![V_b, V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_puntil(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_x)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x)))) | (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) & p(s(t_bool,h4s_pastu_u_temporalu_u_logics_pnext(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_puntil(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_x)))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c14, axiom, ![V_b, V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pwhen(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_x)))) <=> ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x))))) | (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x))))) & p(s(t_bool,h4s_pastu_u_temporalu_u_logics_pnext(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pwhen(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_x)))))))).
fof(ah4s_Pastu_u_Temporalu_u_Logics_RECURSIONu_c15, axiom, ![V_b, V_a, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pbefore(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_x)))) <=> (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_x))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_x)))) | p(s(t_bool,h4s_pastu_u_temporalu_u_logics_pnext(s(t_fun(t_h4s_nums_num,t_bool),h4s_pastu_u_temporalu_u_logics_pbefore(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_x)))))))).
fof(ch4s_Omegau_u_Automatas_TEMPu_u_OPSu_u_DEFSu_u_TOu_u_OMEGAu_c5, conjecture, ![V_l, V_b, V_a]: (s(t_fun(t_h4s_nums_num,t_bool),V_l) = s(t_fun(t_h4s_nums_num,t_bool),h4s_temporalu_u_logics_sbefore(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_fun(t_h4s_nums_num,t_bool),V_b))) <=> (p(s(t_bool,t)) & (![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_l),s(t_h4s_nums_num,V_t)))) <=> (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,V_t))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_a),s(t_h4s_nums_num,V_t)))) | p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_l),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_t))))))))) & ![V_t1]: ?[V_t2]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_l),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t1),s(t_h4s_nums_num,V_t2)))))) => (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_t1),s(t_h4s_nums_num,V_t2)))))) | 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_t1),s(t_h4s_nums_num,V_t2)))))))))))).
