%   ORIGINAL: h4/Temporal__Logic/SWHEN__IMP
% 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/Temporal__Logic/SWHEN0: !t0 b a. h4/Temporal__Logic/SWHEN 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/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)))
% Assm: h4/Temporal__Logic/BEFORE0: !t0 b a. h4/Temporal__Logic/BEFORE 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)) \/ (!t. ~b (h4/arithmetic/_2B t t0))))
% Assm: h4/Temporal__Logic/SBEFORE0: !t0 b a. h4/Temporal__Logic/SBEFORE 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/BEFORE__IMP: !t0 b a. h4/Temporal__Logic/BEFORE 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)) \/ (!t. ~b (h4/arithmetic/_2B t t0)))
% Assm: h4/Temporal__Logic/WHEN__IMP: !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/SUNTIL0: !t0 b a. h4/Temporal__Logic/SUNTIL 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)) /\ (?t. b (h4/arithmetic/_2B t t0)))
% Assm: h4/Temporal__Logic/UNTIL__IMP: !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)))
% Assm: h4/Temporal__Logic/WATCH__EXISTS: !t0 b. ?q. h4/Temporal__Logic/WATCH q b t0
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% 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/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/arithmetic/SUB__EQUAL__0: !c. h4/arithmetic/_2D c c = h4/num/0
% Assm: h4/arithmetic/ADD__SUB: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% 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/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% 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/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/Temporal__Logic/EVENTUAL0: !t0 P. h4/Temporal__Logic/EVENTUAL P t0 <=> (?t. P (h4/arithmetic/_2B t t0))
% Assm: h4/Temporal__Logic/ALWAYS0: !t0 P. h4/Temporal__Logic/ALWAYS 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/arithmetic/MOD__ELIM: !x n P. h4/prim__rec/_3C h4/num/0 n /\ P x /\ (!y. P (h4/arithmetic/_2B y n) ==> P y) ==> P (h4/arithmetic/MOD x n)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/list/DISJOINT__GENLIST__PLUS: !x n b a. h4/pred__set/DISJOINT x (h4/list/LIST__TO__SET (h4/list/GENLIST (h4/arithmetic/_2B n) (h4/arithmetic/_2B a b))) ==> h4/pred__set/DISJOINT x (h4/list/LIST__TO__SET (h4/list/GENLIST (h4/arithmetic/_2B n) a)) /\ h4/pred__set/DISJOINT x (h4/list/LIST__TO__SET (h4/list/GENLIST (h4/arithmetic/_2B (h4/arithmetic/_2B n a)) b))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/pred__set/SUM__SET__THM_c1: !x s. h4/pred__set/FINITE s ==> h4/pred__set/SUM__SET (h4/pred__set/INSERT x s) = h4/arithmetic/_2B x (h4/pred__set/SUM__SET (h4/pred__set/DELETE s x))
% Assm: h4/pred__set/SUM__SET__UNION: !t s. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/SUM__SET (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/SUM__SET s) (h4/pred__set/SUM__SET t)) (h4/pred__set/SUM__SET (h4/pred__set/INTER s t))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/pred__set/MIN__SET__ELIM: !Q P. ~(P = h4/pred__set/EMPTY) /\ (!x. (!y. h4/bool/IN y P ==> h4/arithmetic/_3C_3D x y) /\ h4/bool/IN x P ==> Q x) ==> Q (h4/pred__set/MIN__SET P)
% Assm: h4/while/LEAST__EXISTS: !p. (?n. p n) <=> p (h4/while/LEAST p) /\ (!n. h4/prim__rec/_3C n (h4/while/LEAST p) ==> ~p n)
% Assm: h4/pred__set/MIN__SET__LEM: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/MIN__SET s) s /\ (!x. h4/bool/IN x s ==> h4/arithmetic/_3C_3D (h4/pred__set/MIN__SET s) x)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/arithmetic/WOP: !P. (?n. P n) ==> (?n. P n /\ (!m. h4/prim__rec/_3C m n ==> ~P m))
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% 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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% 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/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% 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/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/pred__set/SUM__SET__DEF: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/COMPLETE__INDUCTION: !P. (!n. (!m. h4/prim__rec/_3C m n ==> P m) ==> P n) ==> (!n. P n)
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/arithmetic/ADD__MODULUS_c0: !x n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MOD (h4/arithmetic/_2B x n) n = h4/arithmetic/MOD x n
% Assm: h4/arithmetic/ADD__MODULUS_c1: !x n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MOD (h4/arithmetic/_2B n x) n = h4/arithmetic/MOD x n
% Assm: h4/arithmetic/SUB__RIGHT__LESS: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2D m n) p <=> h4/prim__rec/_3C m (h4/arithmetic/_2B n p) /\ h4/prim__rec/_3C h4/num/0 p
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/arithmetic/LESS__MOD: !n k. h4/prim__rec/_3C k n ==> h4/arithmetic/MOD k n = k
% Assm: h4/arithmetic/SUB__ADD: !n m. h4/arithmetic/_3C_3D n m ==> h4/arithmetic/_2B (h4/arithmetic/_2D m n) n = m
% Assm: h4/arithmetic/LESS__ADD__NONZERO: !n m. ~(n = h4/num/0) ==> h4/prim__rec/_3C m (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/NOT__ZERO__LT__ZERO: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm: h4/arithmetic/LESS__TRANS: !p n m. h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm: h4/arithmetic/GREATER__OR__EQ: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/arithmetic/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm: h4/arithmetic/GREATER__DEF: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/pred__set/SUM__IMAGE__UNION: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/SUM__IMAGE f (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/SUM__IMAGE f s) (h4/pred__set/SUM__IMAGE f t)) (h4/pred__set/SUM__IMAGE f (h4/pred__set/INTER s t))
% Assm: h4/pred__set/SUM__IMAGE__THM_c1: !s f e. h4/pred__set/FINITE s ==> h4/pred__set/SUM__IMAGE f (h4/pred__set/INSERT e s) = h4/arithmetic/_2B (f e) (h4/pred__set/SUM__IMAGE f (h4/pred__set/DELETE s e))
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/list/GENLIST__PLUS__APPEND: !n2 n1 a. h4/list/APPEND (h4/list/GENLIST (h4/arithmetic/_2B a) n1) (h4/list/GENLIST (h4/arithmetic/_2B (h4/arithmetic/_2B n1 a)) n2) = h4/list/GENLIST (h4/arithmetic/_2B a) (h4/arithmetic/_2B n1 n2)
% Assm: h4/list/LIST__TO__SET__APPEND: !l2 l1. h4/list/LIST__TO__SET (h4/list/APPEND l1 l2) = h4/pred__set/UNION (h4/list/LIST__TO__SET l1) (h4/list/LIST__TO__SET l2)
% Assm: h4/pred__set/DISJOINT__SYM: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/DISJOINT t s
% Assm: h4/pred__set/DISJOINT__UNION__BOTH_c1: !u t s. h4/pred__set/DISJOINT u (h4/pred__set/UNION s t) <=> h4/pred__set/DISJOINT s u /\ h4/pred__set/DISJOINT t u
% Assm: h4/pred__set/MIN__SET__DEF: h4/pred__set/MIN__SET = h4/while/LEAST
% Assm: h4/bool/EQ__TRANS: !z y x. x = y /\ y = z ==> x = z
% Assm: h4/while/LEAST__INTRO: !x P. P x ==> P (h4/while/LEAST P)
% Assm: h4/while/LESS__LEAST: !m P. h4/prim__rec/_3C m (h4/while/LEAST P) ==> ~P m
% Assm: h4/pred__set/SET__CASES: !s. s = h4/pred__set/EMPTY \/ (?x t. s = h4/pred__set/INSERT x t /\ ~h4/bool/IN x t)
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/while/LEAST__ELIM: !Q P. (?n. P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~P m) /\ P n ==> Q n) ==> Q (h4/while/LEAST P)
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/pred__set/MEMBER__NOT__EMPTY: !s. (?x. h4/bool/IN x s) <=> ~(s = h4/pred__set/EMPTY)
% Assm: h4/while/FULL__LEAST__INTRO: !x P. P x ==> P (h4/while/LEAST P) /\ h4/arithmetic/_3C_3D (h4/while/LEAST P) x
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Goal: !t0 b a. h4/Temporal__Logic/SWHEN 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)))
%   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_Temporalu_u_Logics_SWHEN0]: !t0 b a. h4/Temporal__Logic/SWHEN a b t0 <=> (?q. h4/Temporal__Logic/WATCH q b t0 /\ (?t. ~happ q (happ (h4/arithmetic/_2B t) t0) /\ happ b (happ (h4/arithmetic/_2B t) t0) /\ happ a (happ (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 (happ (h4/arithmetic/_2B t) t0)) <=> happ q (happ (h4/arithmetic/_2B t) t0) \/ happ b (happ (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 (happ (h4/arithmetic/_2B t) t0) \/ (happ b (happ (h4/arithmetic/_2B t) t0) ==> happ a (happ (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 (happ (h4/arithmetic/_2B t) t0) \/ happ b (happ (h4/arithmetic/_2B t) t0) \/ happ a (happ (h4/arithmetic/_2B t) t0)))
% Assm [h4s_Temporalu_u_Logics_BEFORE0]: !t0 b a. h4/Temporal__Logic/BEFORE a b t0 <=> (?q. h4/Temporal__Logic/WATCH q b t0 /\ ((?t. ~happ q (happ (h4/arithmetic/_2B t) t0) /\ ~happ b (happ (h4/arithmetic/_2B t) t0) /\ happ a (happ (h4/arithmetic/_2B t) t0)) \/ (!t. ~happ b (happ (h4/arithmetic/_2B t) t0))))
% Assm [h4s_Temporalu_u_Logics_SBEFORE0]: !t0 b a. h4/Temporal__Logic/SBEFORE a b t0 <=> (?q. h4/Temporal__Logic/WATCH q b t0 /\ (?t. ~happ q (happ (h4/arithmetic/_2B t) t0) /\ ~happ b (happ (h4/arithmetic/_2B t) t0) /\ happ a (happ (h4/arithmetic/_2B t) t0)))
% Assm [h4s_Temporalu_u_Logics_BEFOREu_u_IMP]: !t0 b a. h4/Temporal__Logic/BEFORE a b t0 <=> (!q. h4/Temporal__Logic/WATCH q b t0 ==> (?t. ~happ q (happ (h4/arithmetic/_2B t) t0) /\ ~happ b (happ (h4/arithmetic/_2B t) t0) /\ happ a (happ (h4/arithmetic/_2B t) t0)) \/ (!t. ~happ b (happ (h4/arithmetic/_2B t) t0)))
% Assm [h4s_Temporalu_u_Logics_WHENu_u_IMP]: !t0 b a. h4/Temporal__Logic/WHEN a b t0 <=> (!q. h4/Temporal__Logic/WATCH q b t0 ==> (!t. happ q (happ (h4/arithmetic/_2B t) t0) \/ (happ b (happ (h4/arithmetic/_2B t) t0) ==> happ a (happ (h4/arithmetic/_2B t) t0))))
% Assm [h4s_Temporalu_u_Logics_SUNTIL0]: !t0 b a. h4/Temporal__Logic/SUNTIL a b t0 <=> (?q. h4/Temporal__Logic/WATCH q b t0 /\ (!t. happ q (happ (h4/arithmetic/_2B t) t0) \/ happ b (happ (h4/arithmetic/_2B t) t0) \/ happ a (happ (h4/arithmetic/_2B t) t0)) /\ (?t. happ b (happ (h4/arithmetic/_2B t) t0)))
% Assm [h4s_Temporalu_u_Logics_UNTILu_u_IMP]: !t0 b a. h4/Temporal__Logic/UNTIL a b t0 <=> (!q. h4/Temporal__Logic/WATCH q b t0 ==> (!t. happ q (happ (h4/arithmetic/_2B t) t0) \/ happ b (happ (h4/arithmetic/_2B t) t0) \/ happ a (happ (h4/arithmetic/_2B t) t0)))
% Assm [h4s_Temporalu_u_Logics_WATCHu_u_EXISTS]: !t0 b. ?q. h4/Temporal__Logic/WATCH q b t0
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_arithmetics_ADDu_c1]: !n m. happ (h4/arithmetic/_2B (h4/num/SUC m)) n = h4/num/SUC (happ (h4/arithmetic/_2B m) n)
% 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_arithmetics_SUBu_u_EQUALu_u_0]: !c. h4/arithmetic/_2D c c = h4/num/0
% Assm [h4s_arithmetics_ADDu_u_SUB]: !c a. h4/arithmetic/_2D (happ (h4/arithmetic/_2B a) c) c = a
% 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_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. happ (h4/arithmetic/_2B h4/num/0) m = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c2]: !n m. happ (h4/arithmetic/_2B (h4/num/SUC m)) n = h4/num/SUC (happ (h4/arithmetic/_2B m) n)
% 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_Temporalu_u_Logics_EVENTUAL0]: !t0 P. h4/Temporal__Logic/EVENTUAL P t0 <=> (?t. happ P (happ (h4/arithmetic/_2B t) t0))
% Assm [h4s_Temporalu_u_Logics_ALWAYS0]: !t0 P. h4/Temporal__Logic/ALWAYS P t0 <=> (!t. happ P (happ (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 (happ (h4/arithmetic/_2B t) t0)) /\ happ b (happ (h4/arithmetic/_2B d) t0)) \/ (!d. ~happ b (happ (h4/arithmetic/_2B d) t0))
% Assm [h4s_arithmetics_MODu_u_ELIM]: !x n P. h4/prim__rec/_3C h4/num/0 n /\ happ P x /\ (!y. happ P (happ (h4/arithmetic/_2B y) n) ==> happ P y) ==> happ P (h4/arithmetic/MOD x n)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_lists_DISJOINTu_u_GENLISTu_u_PLUS]: !x n b a. h4/pred__set/DISJOINT x (h4/list/LIST__TO__SET (h4/list/GENLIST (h4/arithmetic/_2B n) (happ (h4/arithmetic/_2B a) b))) ==> h4/pred__set/DISJOINT x (h4/list/LIST__TO__SET (h4/list/GENLIST (h4/arithmetic/_2B n) a)) /\ h4/pred__set/DISJOINT x (h4/list/LIST__TO__SET (h4/list/GENLIST (h4/arithmetic/_2B (happ (h4/arithmetic/_2B n) a)) b))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_predu_u_sets_SUMu_u_SETu_u_THMu_c1]: !x s. h4/pred__set/FINITE s ==> happ h4/pred__set/SUM__SET (h4/pred__set/INSERT x s) = happ (h4/arithmetic/_2B x) (happ h4/pred__set/SUM__SET (h4/pred__set/DELETE s x))
% Assm [h4s_predu_u_sets_SUMu_u_SETu_u_UNION]: !t s. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> happ h4/pred__set/SUM__SET (h4/pred__set/UNION s t) = h4/arithmetic/_2D (happ (h4/arithmetic/_2B (happ h4/pred__set/SUM__SET s)) (happ h4/pred__set/SUM__SET t)) (happ h4/pred__set/SUM__SET (h4/pred__set/INTER s t))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_predu_u_sets_MINu_u_SETu_u_ELIM]: !Q P. ~(P = h4/pred__set/EMPTY) /\ (!x. (!y. h4/bool/IN y P ==> h4/arithmetic/_3C_3D x y) /\ h4/bool/IN x P ==> happ Q x) ==> happ Q (happ h4/pred__set/MIN__SET P)
% Assm [h4s_whiles_LEASTu_u_EXISTS]: !p. (?n. happ p n) <=> happ p (happ h4/while/LEAST p) /\ (!n. h4/prim__rec/_3C n (happ h4/while/LEAST p) ==> ~happ p n)
% Assm [h4s_predu_u_sets_MINu_u_SETu_u_LEM]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (happ h4/pred__set/MIN__SET s) s /\ (!x. h4/bool/IN x s ==> h4/arithmetic/_3C_3D (happ h4/pred__set/MIN__SET s) x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_arithmetics_WOP]: !P. (?n. happ P n) ==> (?n. happ P n /\ (!m. h4/prim__rec/_3C m n ==> ~happ P m))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% 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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% 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_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% 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_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_predu_u_sets_SUMu_u_SETu_u_DEF]: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. happ (h4/arithmetic/_2B m) n = happ (h4/arithmetic/_2B n) m
% Assm [h4s_arithmetics_COMPLETEu_u_INDUCTION]: !P. (!n. (!m. h4/prim__rec/_3C m n ==> happ P m) ==> happ P n) ==> (!n. happ P n)
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_arithmetics_ADDu_u_MODULUSu_c0]: !x n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MOD (happ (h4/arithmetic/_2B x) n) n = h4/arithmetic/MOD x n
% Assm [h4s_arithmetics_ADDu_u_MODULUSu_c1]: !x n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MOD (happ (h4/arithmetic/_2B n) x) n = h4/arithmetic/MOD x n
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_LESS]: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2D m n) p <=> h4/prim__rec/_3C m (happ (h4/arithmetic/_2B n) p) /\ h4/prim__rec/_3C h4/num/0 p
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_arithmetics_LESSu_u_MOD]: !n k. h4/prim__rec/_3C k n ==> h4/arithmetic/MOD k n = k
% Assm [h4s_arithmetics_SUBu_u_ADD]: !n m. h4/arithmetic/_3C_3D n m ==> happ (h4/arithmetic/_2B (h4/arithmetic/_2D m n)) n = m
% Assm [h4s_arithmetics_LESSu_u_ADDu_u_NONZERO]: !n m. ~(n = h4/num/0) ==> h4/prim__rec/_3C m (happ (h4/arithmetic/_2B m) n)
% Assm [h4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO]: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm [h4s_arithmetics_LESSu_u_TRANS]: !p n m. h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm [h4s_arithmetics_GREATERu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_arithmetics_LESSu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm [h4s_arithmetics_GREATERu_u_DEF]: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_predu_u_sets_SUMu_u_IMAGEu_u_UNION]: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> happ (h4/pred__set/SUM__IMAGE f) (h4/pred__set/UNION s t) = h4/arithmetic/_2D (happ (h4/arithmetic/_2B (happ (h4/pred__set/SUM__IMAGE f) s)) (happ (h4/pred__set/SUM__IMAGE f) t)) (happ (h4/pred__set/SUM__IMAGE f) (h4/pred__set/INTER s t))
% Assm [h4s_predu_u_sets_SUMu_u_IMAGEu_u_THMu_c1]: !s f e. h4/pred__set/FINITE s ==> happ (h4/pred__set/SUM__IMAGE f) (h4/pred__set/INSERT e s) = happ (h4/arithmetic/_2B (happ f e)) (happ (h4/pred__set/SUM__IMAGE f) (h4/pred__set/DELETE s e))
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. happ (h4/arithmetic/_2B m) p = happ (h4/arithmetic/_2B n) p <=> m = n
% Assm [h4s_lists_GENLISTu_u_PLUSu_u_APPEND]: !n2 n1 a. h4/list/APPEND (h4/list/GENLIST (h4/arithmetic/_2B a) n1) (h4/list/GENLIST (h4/arithmetic/_2B (happ (h4/arithmetic/_2B n1) a)) n2) = h4/list/GENLIST (h4/arithmetic/_2B a) (happ (h4/arithmetic/_2B n1) n2)
% Assm [h4s_lists_LISTu_u_TOu_u_SETu_u_APPEND]: !l2 l1. h4/list/LIST__TO__SET (h4/list/APPEND l1 l2) = h4/pred__set/UNION (h4/list/LIST__TO__SET l1) (h4/list/LIST__TO__SET l2)
% Assm [h4s_predu_u_sets_DISJOINTu_u_SYM]: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/DISJOINT t s
% Assm [h4s_predu_u_sets_DISJOINTu_u_UNIONu_u_BOTHu_c1]: !u t s. h4/pred__set/DISJOINT u (h4/pred__set/UNION s t) <=> h4/pred__set/DISJOINT s u /\ h4/pred__set/DISJOINT t u
% Assm [h4s_predu_u_sets_MINu_u_SETu_u_DEF]: h4/pred__set/MIN__SET = h4/while/LEAST
% Assm [h4s_bools_EQu_u_TRANS]: !z y x. x = y /\ y = z ==> x = z
% Assm [h4s_whiles_LEASTu_u_INTRO]: !x P. happ P x ==> happ P (happ h4/while/LEAST P)
% Assm [h4s_whiles_LESSu_u_LEAST]: !m P. h4/prim__rec/_3C m (happ h4/while/LEAST P) ==> ~happ P m
% Assm [h4s_predu_u_sets_SETu_u_CASES]: !s. s = h4/pred__set/EMPTY \/ (?x t. s = h4/pred__set/INSERT x t /\ ~h4/bool/IN x t)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_whiles_LEASTu_u_ELIM]: !Q P. (?n. happ P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~happ P m) /\ happ P n ==> happ Q n) ==> happ Q (happ h4/while/LEAST P)
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_predu_u_sets_MEMBERu_u_NOTu_u_EMPTY]: !s. (?x. h4/bool/IN x s) <=> ~(s = h4/pred__set/EMPTY)
% Assm [h4s_whiles_FULLu_u_LEASTu_u_INTRO]: !x P. happ P x ==> happ P (happ h4/while/LEAST P) /\ h4/arithmetic/_3C_3D (happ h4/while/LEAST P) x
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Goal: !t0 b a. h4/Temporal__Logic/SWHEN a b t0 <=> (!q. h4/Temporal__Logic/WATCH q b t0 ==> (?t. ~happ q (happ (h4/arithmetic/_2B t) t0) /\ happ b (happ (h4/arithmetic/_2B t) t0) /\ happ a (happ (h4/arithmetic/_2B t) t0)))
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_Q1395165,TV_Q1395161]: ![V_f, V_g]: (![V_x]: s(TV_Q1395161,happ(s(t_fun(TV_Q1395165,TV_Q1395161),V_f),s(TV_Q1395165,V_x))) = s(TV_Q1395161,happ(s(t_fun(TV_Q1395165,TV_Q1395161),V_g),s(TV_Q1395165,V_x))) => s(t_fun(TV_Q1395165,TV_Q1395161),V_f) = s(t_fun(TV_Q1395165,TV_Q1395161),V_g))).
fof(ah4s_Temporalu_u_Logics_SWHEN0, axiom, ![V_t0, V_b, V_a]: (p(s(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_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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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_BEFORE0, axiom, ![V_t0, V_b, V_a]: (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_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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t))),s(t_h4s_nums_num,V_t0)))))))) | ![V_t]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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_SBEFORE0, axiom, ![V_t0, V_b, V_a]: (p(s(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_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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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_BEFOREu_u_IMP, axiom, ![V_t0, V_b, V_a]: (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_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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t))),s(t_h4s_nums_num,V_t0)))))))) | ![V_t]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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_WHENu_u_IMP, 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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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_SUNTIL0, axiom, ![V_t0, V_b, V_a]: (p(s(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_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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t))),s(t_h4s_nums_num,V_t0)))))))) & ?[V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_b),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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_UNTILu_u_IMP, 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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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_WATCHu_u_EXISTS, axiom, ![V_t0, V_b]: ?[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))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t1))).
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,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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_ADDu_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))).
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_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_arithmetics_ADDu_u_SUB, axiom, ![V_c, V_a]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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_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_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t1) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => 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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))).
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_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,happ(s(t_fun(t_h4s_nums_num,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_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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_d))),s(t_h4s_nums_num,V_t0))))))))).
fof(ah4s_arithmetics_MODu_u_ELIM, axiom, ![V_x, V_n, V_P]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),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,V_x)))) & ![V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_y))),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,V_y))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_bools_IMPu_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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t1)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t1))) <=> p(s(t_bool,t1)))).
fof(ah4s_lists_DISJOINTu_u_GENLISTu_u_PLUS, axiom, ![V_x, V_n, V_b, V_a]: (p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(t_h4s_nums_num,t_bool),V_x),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_b)))))))))) => (p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(t_h4s_nums_num,t_bool),V_x),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_a)))))))) & p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(t_h4s_nums_num,t_bool),V_x),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_a))))),s(t_h4s_nums_num,V_b))))))))))).
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_predu_u_sets_SUMu_u_SETu_u_THMu_c1, axiom, ![V_x, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_delete(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_h4s_nums_num,V_x))))))))).
fof(ah4s_predu_u_sets_SUMu_u_SETu_u_UNION, axiom, ![V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_t))))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_bool),V_t))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_s))))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_t))))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_inter(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_bool),V_t))))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t1))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_MINu_u_SETu_u_ELIM, axiom, ![V_Q, V_P]: ((~ (s(t_fun(t_h4s_nums_num,t_bool),V_P) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)) & ![V_x]: ((![V_y]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_y),s(t_fun(t_h4s_nums_num,t_bool),V_P)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_y))))) & p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),V_P))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,V_x)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_whiles_LEASTu_u_EXISTS, axiom, ![V_p]: (?[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,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_p)))))) & ![V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_p)))))) => ~ (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_predu_u_sets_MINu_u_SETu_u_LEM, axiom, ![V_s]: (~ (s(t_fun(t_h4s_nums_num,t_bool),V_s) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)) => (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_s))),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_s))),s(t_h4s_nums_num,V_x)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t1))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t1) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t1)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t1)))).
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_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_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
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,t1)))).
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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t1) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_WOP, axiom, ![V_P]: (?[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)))) => ?[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)))) & ![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,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))))).
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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_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_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_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_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_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_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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_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_predu_u_sets_SUMu_u_SETu_u_DEF, axiom, s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t1))) <=> p(s(t_bool,t1)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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_COMPLETEu_u_INDUCTION, axiom, ![V_P]: (![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,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),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_bools_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
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_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_arithmetics_ADDu_u_MODULUSu_c0, axiom, ![V_x, 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,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_ADDu_u_MODULUSu_c1, axiom, ![V_x, 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,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_LESS, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_p))))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_arithmetics_LESSu_u_MOD, axiom, ![V_n, V_k]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_k))).
fof(ah4s_arithmetics_SUBu_u_ADD, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) => s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_m))).
fof(ah4s_arithmetics_LESSu_u_ADDu_u_NONZERO, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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_NOTu_u_ZEROu_u_LTu_u_ZERO, axiom, ![V_n]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_LESSu_u_TRANS, axiom, ![V_p, 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_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_GREATERu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) <=> (p(s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_arithmetics_LESSu_u_ORu_u_EQ, 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)))) <=> (p(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_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_GREATERu_u_DEF, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_predu_u_sets_SUMu_u_IMAGEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),V_t))))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))))))).
fof(ah4s_predu_u_sets_SUMu_u_IMAGEu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_s, V_f, V_e]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_e))))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_e))))))))).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_lists_GENLISTu_u_PLUSu_u_APPEND, axiom, ![V_n2, V_n1, V_a]: s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_append(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_n1))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n1))),s(t_h4s_nums_num,V_a))))),s(t_h4s_nums_num,V_n2))))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n1))),s(t_h4s_nums_num,V_n2)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SETu_u_APPEND, axiom, ![TV_u_27a]: ![V_l2, V_l1]: s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_predu_u_sets_DISJOINTu_u_SYM, axiom, ![TV_u_27a]: ![V_t, V_s]: s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_predu_u_sets_DISJOINTu_u_UNIONu_u_BOTHu_c1, axiom, ![TV_u_27a]: ![V_u, V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_u),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_u)))) & p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))))))).
fof(ah4s_predu_u_sets_MINu_u_SETu_u_DEF, axiom, s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least)).
fof(ah4s_bools_EQu_u_TRANS, axiom, ![TV_u_27a]: ![V_z, 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_z)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_z))).
fof(ah4s_whiles_LEASTu_u_INTRO, axiom, ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_x)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_whiles_LESSu_u_LEAST, axiom, ![V_m, V_P]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_P)))))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_predu_u_sets_SETu_u_CASES, axiom, ![TV_u_27a]: ![V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty) | ?[V_x, V_t]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))) & ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_whiles_LEASTu_u_ELIM, axiom, ![V_Q, V_P]: ((?[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)))) & ![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,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m)))))) & 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_Q),s(t_h4s_nums_num,V_n)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_bools_CONJu_u_ASSOC, 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_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> ?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
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_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_INu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_x0]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_x0))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x0),s(TV_u_27a,V_x)))).
fof(ah4s_predu_u_sets_MEMBERu_u_NOTu_u_EMPTY, axiom, ![TV_u_27a]: ![V_s]: (?[V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> ~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))).
fof(ah4s_whiles_FULLu_u_LEASTu_u_INTRO, axiom, ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_x)))) => (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_P)))))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_P))),s(t_h4s_nums_num,V_x))))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ch4s_Temporalu_u_Logics_SWHENu_u_IMP, conjecture, ![V_t0, V_b, V_a]: (p(s(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_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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_t))),s(t_h4s_nums_num,V_t0))))))))))).
