%   ORIGINAL: h4/path/okpath__thm_c0
% 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/path/okpath__def: !R. h4/path/okpath R = h4/fixedPoint/gfp (h4/path/okpath__f R)
% Assm: h4/path/okpath__monotone: !R. h4/fixedPoint/monotone (h4/path/okpath__f R)
% Assm: h4/path/okpath__cases: !x R. h4/path/okpath R x <=> (?x_27. x = h4/path/stopped__at x_27) \/ (?x_27 r p. x = h4/path/pcons x_27 r p /\ R x_27 r (h4/path/first p) /\ h4/path/okpath R p)
% Assm: h4/path/okpath__co__ind: !R P. (!x r p. P (h4/path/pcons x r p) ==> R x r (h4/path/first p) /\ P p) ==> (!p. P p ==> h4/path/okpath R p)
% Assm: h4/bool/TRUTH: T
% Assm: h4/path/PL__stopped__at: !x. h4/path/PL (h4/path/stopped__at x) = h4/pred__set/INSERT h4/num/0 h4/pred__set/EMPTY
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/path/stopped__at__11: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm: h4/path/exists__thm_c0: !x P. h4/path/exists P (h4/path/stopped__at x) <=> P x
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/path/okpath__f__def: !X R. h4/path/okpath__f R X = h4/pred__set/UNION (h4/pred__set/GSPEC (\x. h4/pair/_2C (h4/path/stopped__at x) (h4/bool/IN x h4/pred__set/UNIV))) (h4/pred__set/GSPEC (h4/pair/UNCURRY (\x. h4/pair/UNCURRY (\r p. h4/pair/_2C (h4/path/pcons x r p) (R x r (h4/path/first p) /\ h4/bool/IN p X)))))
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/path/PL__thm_c0: !x. h4/path/PL (h4/path/stopped__at x) = h4/pred__set/INSERT h4/num/0 h4/pred__set/EMPTY
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/path/firstP__at__thm_c0: !x n P. h4/path/firstP__at P (h4/path/stopped__at x) n <=> n = h4/num/0 /\ P x
% Assm: h4/path/stopped__at__def: !x. h4/path/stopped__at x = h4/path/toPath (h4/pair/_2C x h4/llist/LNIL)
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/path/length__thm_c0: !x. h4/path/length (h4/path/stopped__at x) = h4/option/SOME (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/path/every__thm_c0: !x P. h4/path/every P (h4/path/stopped__at x) <=> P x
% Assm: h4/path/alt__length__thm_c0: !x. h4/path/length (h4/path/stopped__at x) = h4/option/SOME (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/path/is__stopped__thm_c0: !x. h4/path/is__stopped (h4/path/stopped__at x) <=> T
% Assm: h4/path/mem__thm_c0: !x s. h4/path/mem s (h4/path/stopped__at x) <=> s = x
% Assm: h4/path/stopped__at__not__pcons_c1: !y x r p. ~(h4/path/pcons y r p = h4/path/stopped__at x)
% Assm: h4/path/filter__def_c0: !x P. P x ==> h4/path/filter P (h4/path/stopped__at x) = h4/path/stopped__at x
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/path/toPath__11: !r_27 r. h4/path/toPath r = h4/path/toPath r_27 <=> r = r_27
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/fixedPoint/gfp__greatest__fixedpoint: !f. h4/fixedPoint/monotone f ==> h4/fixedPoint/gfp f = f (h4/fixedPoint/gfp f) /\ (!X. X = f X ==> h4/pred__set/SUBSET X (h4/fixedPoint/gfp f))
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/path/every__def: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Assm: h4/path/is__stopped__def: !p. h4/path/is__stopped p <=> (?x. p = h4/path/stopped__at x)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/path/pcons__11: !y x s r q p. h4/path/pcons x r p = h4/path/pcons y s q <=> x = y /\ r = s /\ p = q
% Assm: h4/path/stopped__at__not__pcons_c0: !y x r p. ~(h4/path/stopped__at x = h4/path/pcons y r p)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/fixedPoint/monotone__def: !f. h4/fixedPoint/monotone f <=> (!X Y. h4/pred__set/SUBSET X Y ==> h4/pred__set/SUBSET (f X) (f Y))
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% 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/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/path/first__thm_c0: !x. h4/path/first (h4/path/stopped__at x) = x
% Assm: h4/path/el__def_c0: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/path/FORALL__path: !P. (!p. P p) <=> (!x. P (h4/path/stopped__at x)) /\ (!x r p. P (h4/path/pcons x r p))
% Assm: h4/fixedPoint/gfp__coinduction: !f. h4/fixedPoint/monotone f ==> (!X. h4/pred__set/SUBSET X (f X) ==> h4/pred__set/SUBSET X (h4/fixedPoint/gfp f))
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/llist/toList__THM_c0: h4/llist/toList h4/llist/LNIL = h4/option/SOME h4/list/NIL
% Assm: h4/path/path__rep__bijections__thm_c1: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/path/length__def: !p. h4/path/length p = h4/bool/COND (h4/path/finite p) (h4/option/SOME (h4/arithmetic/_2B (h4/list/LENGTH (h4/option/THE (h4/llist/toList (h4/pair/SND (h4/path/fromPath p))))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) h4/option/NONE
% Assm: h4/path/finite__thm_c0: !x. h4/path/finite (h4/path/stopped__at x) <=> T
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/path/pcons__def: !x r p. h4/path/pcons x r p = h4/path/toPath (h4/pair/_2C x (h4/llist/LCONS (h4/pair/_2C r (h4/path/first p)) (h4/pair/SND (h4/path/fromPath p))))
% Assm: h4/llist/LCONS__NOT__NIL_c0: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm: h4/path/mem__def: !s p. h4/path/mem s p <=> (?i. h4/bool/IN i (h4/path/PL p) /\ s = h4/path/el i p)
% Assm: h4/path/exists__def: !p P. h4/path/exists P p <=> (?i. h4/path/firstP__at P p i)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/path/firstP__at__def: !p i P. h4/path/firstP__at P p i <=> h4/bool/IN i (h4/path/PL p) /\ P (h4/path/el i p) /\ (!j. h4/prim__rec/_3C j i ==> ~P (h4/path/el j p))
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/bool/SELECT__REFL: !x. h4/min/_40 (\y. y = x) = x
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% 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_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/arithmetic/SUB__RIGHT__EQ: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Goal: !x R. h4/path/okpath R (h4/path/stopped__at x)
%   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_paths_okpathu_u_def]: !R. h4/path/okpath R = h4/fixedPoint/gfp (h4/path/okpath__f R)
% Assm [h4s_paths_okpathu_u_monotone]: !R. h4/fixedPoint/monotone (h4/path/okpath__f R)
% Assm [h4s_paths_okpathu_u_cases]: !x R. happ (h4/path/okpath R) x <=> (?x_27. x = h4/path/stopped__at x_27) \/ (?x_27 r p. x = h4/path/pcons x_27 r p /\ happ (happ (happ R x_27) r) (h4/path/first p) /\ happ (h4/path/okpath R) p)
% Assm [h4s_paths_okpathu_u_cou_u_ind]: !R P. (!x r p. happ P (h4/path/pcons x r p) ==> happ (happ (happ R x) r) (h4/path/first p) /\ happ P p) ==> (!p. happ P p ==> happ (h4/path/okpath R) p)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_paths_PLu_u_stoppedu_u_at]: !x. h4/path/PL (h4/path/stopped__at x) = h4/pred__set/INSERT h4/num/0 h4/pred__set/EMPTY
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_paths_stoppedu_u_atu_u_11]: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm [h4s_paths_existsu_u_thmu_c0]: !x P. h4/path/exists P (h4/path/stopped__at x) <=> happ P x
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_paths_okpathu_u_fu_u_def]: !_3. (!R x r X p. ?v. (v <=> happ (happ (happ R x) r) (h4/path/first p) /\ h4/bool/IN p X) /\ happ (happ (happ (happ (happ _3 R) x) r) X) p = h4/pair/_2C (h4/path/pcons x r p) v) ==> (!_2. (!R x X r. happ (happ (happ (happ _2 R) x) X) r = happ (happ (happ (happ _3 R) x) r) X) ==> (!_1. (!R X x. happ (happ (happ _1 R) X) x = h4/pair/UNCURRY (happ (happ (happ _2 R) x) X)) ==> (!_0. (!x. happ _0 x = h4/pair/_2C (h4/path/stopped__at x) (h4/bool/IN x h4/pred__set/UNIV)) ==> (!X R. happ (h4/path/okpath__f R) X = h4/pred__set/UNION (h4/pred__set/GSPEC _0) (h4/pred__set/GSPEC (h4/pair/UNCURRY (happ (happ _1 R) X)))))))
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_paths_PLu_u_thmu_c0]: !x. h4/path/PL (h4/path/stopped__at x) = h4/pred__set/INSERT h4/num/0 h4/pred__set/EMPTY
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_paths_firstPu_u_atu_u_thmu_c0]: !x n P. h4/path/firstP__at P (h4/path/stopped__at x) n <=> n = h4/num/0 /\ happ P x
% Assm [h4s_paths_stoppedu_u_atu_u_def]: !x. h4/path/stopped__at x = h4/path/toPath (h4/pair/_2C x h4/llist/LNIL)
% 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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_paths_lengthu_u_thmu_c0]: !x. h4/path/length (h4/path/stopped__at x) = h4/option/SOME (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_paths_everyu_u_thmu_c0]: !x P. h4/path/every P (h4/path/stopped__at x) <=> happ P x
% Assm [h4s_paths_altu_u_lengthu_u_thmu_c0]: !x. h4/path/length (h4/path/stopped__at x) = h4/option/SOME (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_paths_isu_u_stoppedu_u_thmu_c0]: !x. h4/path/is__stopped (h4/path/stopped__at x) <=> T
% Assm [h4s_paths_memu_u_thmu_c0]: !x s. h4/path/mem s (h4/path/stopped__at x) <=> s = x
% Assm [h4s_paths_stoppedu_u_atu_u_notu_u_pconsu_c1]: !y x r p. ~(h4/path/pcons y r p = h4/path/stopped__at x)
% Assm [h4s_paths_filteru_u_defu_c0]: !x P. happ P x ==> h4/path/filter P (h4/path/stopped__at x) = h4/path/stopped__at x
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_paths_toPathu_u_11]: !r_27 r. h4/path/toPath r = h4/path/toPath r_27 <=> r = r_27
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_fixedPoints_gfpu_u_greatestu_u_fixedpoint]: !f. h4/fixedPoint/monotone f ==> h4/fixedPoint/gfp f = happ f (h4/fixedPoint/gfp f) /\ (!X. X = happ f X ==> h4/pred__set/SUBSET X (h4/fixedPoint/gfp f))
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_paths_everyu_u_def]: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Assm [h4s_paths_isu_u_stoppedu_u_def]: !p. h4/path/is__stopped p <=> (?x. p = h4/path/stopped__at x)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_paths_pconsu_u_11]: !y x s r q p. h4/path/pcons x r p = h4/path/pcons y s q <=> x = y /\ r = s /\ p = q
% Assm [h4s_paths_stoppedu_u_atu_u_notu_u_pconsu_c0]: !y x r p. ~(h4/path/stopped__at x = h4/path/pcons y r p)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_fixedPoints_monotoneu_u_def]: !f. h4/fixedPoint/monotone f <=> (!X Y. h4/pred__set/SUBSET X Y ==> h4/pred__set/SUBSET (happ f X) (happ f Y))
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% 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_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_paths_firstu_u_thmu_c0]: !x. h4/path/first (h4/path/stopped__at x) = x
% Assm [h4s_paths_elu_u_defu_c0]: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_paths_FORALLu_u_path]: !P. (!p. happ P p) <=> (!x. happ P (h4/path/stopped__at x)) /\ (!x r p. happ P (h4/path/pcons x r p))
% Assm [h4s_fixedPoints_gfpu_u_coinduction]: !f. h4/fixedPoint/monotone f ==> (!X. h4/pred__set/SUBSET X (happ f X) ==> h4/pred__set/SUBSET X (h4/fixedPoint/gfp f))
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_llists_toListu_u_THMu_c0]: h4/llist/toList h4/llist/LNIL = h4/option/SOME h4/list/NIL
% Assm [h4s_paths_pathu_u_repu_u_bijectionsu_u_thmu_c1]: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (h4/option/SOME x) = x
% Assm [h4s_paths_lengthu_u_def]: !p. h4/path/length p = h4/bool/COND (h4/path/finite p) (h4/option/SOME (h4/arithmetic/_2B (h4/list/LENGTH (h4/option/THE (h4/llist/toList (h4/pair/SND (h4/path/fromPath p))))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) h4/option/NONE
% Assm [h4s_paths_finiteu_u_thmu_c0]: !x. h4/path/finite (h4/path/stopped__at x) <=> T
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_paths_pconsu_u_def]: !x r p. h4/path/pcons x r p = h4/path/toPath (h4/pair/_2C x (h4/llist/LCONS (h4/pair/_2C r (h4/path/first p)) (h4/pair/SND (h4/path/fromPath p))))
% Assm [h4s_llists_LCONSu_u_NOTu_u_NILu_c0]: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm [h4s_paths_memu_u_def]: !s p. h4/path/mem s p <=> (?i. h4/bool/IN i (h4/path/PL p) /\ s = h4/path/el i p)
% Assm [h4s_paths_existsu_u_def]: !p P. h4/path/exists P p <=> (?i. h4/path/firstP__at P p i)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_paths_firstPu_u_atu_u_def]: !p i P. h4/path/firstP__at P p i <=> h4/bool/IN i (h4/path/PL p) /\ happ P (h4/path/el i p) /\ (!j. h4/prim__rec/_3C j i ==> ~happ P (h4/path/el j p))
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_bools_SELECTu_u_REFL]: !_0. (!x y. happ (happ _0 x) y <=> y = x) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% 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_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_EQ]: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Goal: !x R. happ (h4/path/okpath R) (h4/path/stopped__at x)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1230102,TV_Q1230098]: ![V_f, V_g]: (![V_x]: s(TV_Q1230098,happ(s(t_fun(TV_Q1230102,TV_Q1230098),V_f),s(TV_Q1230102,V_x))) = s(TV_Q1230098,happ(s(t_fun(TV_Q1230102,TV_Q1230098),V_g),s(TV_Q1230102,V_x))) => s(t_fun(TV_Q1230102,TV_Q1230098),V_f) = s(t_fun(TV_Q1230102,TV_Q1230098),V_g))).
fof(ah4s_paths_okpathu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_R]: s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_paths_okpath(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))) = s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_fixedpoints_gfp(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),h4s_paths_okpathu_u_f(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R)))))).
fof(ah4s_paths_okpathu_u_monotone, axiom, ![TV_u_27b,TV_u_27a]: ![V_R]: p(s(t_bool,h4s_fixedpoints_monotone(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),h4s_paths_okpathu_u_f(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))))))).
fof(ah4s_paths_okpathu_u_cases, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_paths_okpath(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_x)))) <=> (?[V_xu_27]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_xu_27))) | ?[V_xu_27, V_r, V_p]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_xu_27),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R),s(TV_u_27a,V_xu_27))),s(TV_u_27b,V_r))),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_paths_okpath(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))))).
fof(ah4s_paths_okpathu_u_cou_u_ind, axiom, ![TV_u_27a,TV_u_27b]: ![V_R, V_P]: (![V_x, V_r, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_r))),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) => ![V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) => p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_paths_okpath(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_paths_PLu_u_stoppedu_u_at, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,h4s_nums_0),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)))).
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_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_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,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_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_paths_stoppedu_u_atu_u_11, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_paths_existsu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, 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_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_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | 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_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_paths_okpathu_u_fu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_3]: (![V_R, V_x, V_r, V_X, V_p]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_r))),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & p(s(t_bool,h4s_bools_in(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_X)))))) & s(t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(TV_u_27a,V_x))),s(TV_u_27b,V_r))),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_X))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_u_2c(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))),s(t_bool,V_v)))) => ![V_uu_2]: (![V_R, V_x, V_X, V_r]: s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27b,t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27b,t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27b,t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_X))),s(TV_u_27b,V_r))) = s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(TV_u_27a,V_x))),s(TV_u_27b,V_r))),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_X))) => ![V_uu_1]: (![V_R, V_X, V_x]: s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b)),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b)),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b)),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b)),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_X))),s(TV_u_27a,V_x))) = s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b)),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27b,t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27b,t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27b,t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27b,t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_X))))) => ![V_uu_0]: (![V_x]: s(t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_u_2c(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))),s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))) => ![V_X, V_R]: s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),h4s_paths_okpathu_u_f(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_X))) = s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),V_uu_0))),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b))),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b)),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b)),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),t_fun(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_paths_path(TV_u_27a,TV_u_27b)),t_h4s_pairs_prod(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_X)))))))))))))).
fof(ah4s_predu_u_sets_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: 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_univ))))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_paths_PLu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,h4s_nums_0),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_paths_firstPu_u_atu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_n, V_P]: (p(s(t_bool,h4s_paths_firstpu_u_at(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n)))) <=> (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_paths_stoppedu_u_atu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_llists_lnil)))))).
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_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_paths_lengthu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_paths_length(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(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_paths_everyu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_paths_altu_u_lengthu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_paths_length(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(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_paths_isu_u_stoppedu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_bool,h4s_paths_isu_u_stopped(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_paths_memu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_s]: (p(s(t_bool,h4s_paths_mem(s(TV_u_27a,V_s),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x)))))) <=> s(TV_u_27a,V_s) = s(TV_u_27a,V_x))).
fof(ah4s_paths_stoppedu_u_atu_u_notu_u_pconsu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_r, V_p]: ~ (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_y),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))).
fof(ah4s_paths_filteru_u_defu_c0, axiom, ![TV_u_27b,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)))) => s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
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_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_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & 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,V_a)))))).
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_paths_toPathu_u_11, axiom, ![TV_u_27b,TV_u_27a]: ![V_ru_27, V_r]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_r))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_ru_27))) <=> s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_r) = s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_ru_27))).
fof(ah4s_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_fixedPoints_gfpu_u_greatestu_u_fixedpoint, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_fixedpoints_monotone(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f)))) => (s(t_fun(TV_u_27a,t_bool),h4s_fixedpoints_gfp(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_fixedpoints_gfp(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f))))) & ![V_X]: (s(t_fun(TV_u_27a,t_bool),V_X) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(TV_u_27a,t_bool),V_X))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_X),s(t_fun(TV_u_27a,t_bool),h4s_fixedpoints_gfp(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f)))))))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = 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_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_predu_u_sets_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![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)))) => 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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_paths_everyu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (p(s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ~ (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))).
fof(ah4s_paths_isu_u_stoppedu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_paths_isu_u_stopped(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ?[V_x]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))).
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_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_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_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_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_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_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_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_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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(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_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_paths_pconsu_u_11, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_s, V_r, V_q, V_p]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_y),s(TV_u_27b,V_s),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) & (s(TV_u_27b,V_r) = s(TV_u_27b,V_s) & s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))).
fof(ah4s_paths_stoppedu_u_atu_u_notu_u_pconsu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_r, V_p]: ~ (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_y),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))).
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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_fixedPoints_monotoneu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_fixedpoints_monotone(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_f)))) <=> ![V_X, V_Y]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_X),s(t_fun(TV_u_27a,t_bool),V_Y)))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_f),s(t_fun(TV_u_27a,t_bool),V_X))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),V_f),s(t_fun(TV_u_27a,t_bool),V_Y))))))))).
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_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_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
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_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (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_empty)))))).
fof(ah4s_paths_firstu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_paths_elu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))).
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_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_0, 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_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_paths_FORALLu_u_path, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x)))))) & ![V_x, V_r, V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))))).
fof(ah4s_fixedPoints_gfpu_u_coinduction, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_fixedpoints_monotone(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f)))) => ![V_X]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_X),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(TV_u_27a,t_bool),V_X)))))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_X),s(t_fun(TV_u_27a,t_bool),h4s_fixedpoints_gfp(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_f))))))))).
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_llists_toListu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_tolist(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_some(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_paths_pathu_u_repu_u_bijectionsu_u_thmu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_r]: s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_paths_frompath(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_r))))) = s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_r)).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_THEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_paths_lengthu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(t_h4s_options_option(t_h4s_nums_num),h4s_paths_length(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_options_the(s(t_h4s_options_option(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_llists_tolist(s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_paths_frompath(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),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))))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)))).
fof(ah4s_paths_finiteu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
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_paths_pconsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_llists_lcons(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_r),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_paths_frompath(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))))))).
fof(ah4s_llists_LCONSu_u_NOTu_u_NILu_c0, axiom, ![TV_u_27a]: ![V_t, V_h]: ~ (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))).
fof(ah4s_paths_memu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_p]: (p(s(t_bool,h4s_paths_mem(s(TV_u_27a,V_s),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ?[V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & s(TV_u_27a,V_s) = s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_existsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ?[V_i]: p(s(t_bool,h4s_paths_firstpu_u_at(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p),s(t_h4s_nums_num,V_i)))))).
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_primu_u_recs_NOTu_u_LESSu_u_0, 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,h4s_nums_0)))))).
fof(ah4s_paths_firstPu_u_atu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_i, V_P]: (p(s(t_bool,h4s_paths_firstpu_u_at(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p),s(t_h4s_nums_num,V_i)))) <=> (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & ![V_j]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_i)))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_j),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))))))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, 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_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
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_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,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))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
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_numerals_numeralu_u_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(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_bools_SELECTu_u_REFL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
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_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(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_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_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_c17, axiom, ![V_n]: (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_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_EQ, axiom, ![V_p, V_n, V_m]: (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) <=> (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))) | (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_p),s(t_h4s_nums_num,h4s_nums_0)))))))).
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_arithmetics_NOTu_u_LEQ, 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_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(ch4s_paths_okpathu_u_thmu_c0, conjecture, ![TV_u_27b,TV_u_27a]: ![V_x, V_R]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),h4s_paths_okpath(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))))).
