%   ORIGINAL: h4/path/okpath__co__ind
% 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__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/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/path/okpath__def: !R. h4/path/okpath R = h4/fixedPoint/gfp (h4/path/okpath__f R)
% Assm: h4/path/every__coinduction: !Q P. (!x. P (h4/path/stopped__at x) ==> Q x) /\ (!x r p. P (h4/path/pcons x r p) ==> Q x /\ P p) ==> (!p. P p ==> h4/path/every Q p)
% Assm: h4/path/take__def_c1: !p n. h4/path/take (h4/num/SUC n) p = h4/path/pcons (h4/path/first p) (h4/path/first__label p) (h4/path/take n (h4/path/tail p))
% 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/path/take__def_c0: !p. h4/path/take h4/num/0 p = h4/path/stopped__at (h4/path/first p)
% Assm: h4/path/EXISTS__path: !P. (?p. P p) <=> (?x. P (h4/path/stopped__at x)) \/ (?x r p. P (h4/path/pcons x r p))
% Assm: h4/path/finite__path__ind: !P. (!x. P (h4/path/stopped__at x)) /\ (!x r p. h4/path/finite p /\ P p ==> P (h4/path/pcons x r p)) ==> (!q. h4/path/finite q ==> P q)
% Assm: h4/path/exists__induction: !Q P. (!x. Q x ==> P (h4/path/stopped__at x)) /\ (!x r p. Q x ==> P (h4/path/pcons x r p)) /\ (!x r p. P p ==> P (h4/path/pcons x r p)) ==> (!p. h4/path/exists Q p ==> P p)
% Assm: h4/bool/TRUTH: T
% Assm: h4/arithmetic/SUC__ELIM__NUMERALS: !g f. (!n. g (h4/num/SUC n) = f n (h4/num/SUC n)) <=> (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/FORALL__SIMP: !t. (!x. 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/REFL__CLAUSE: !x. x = x <=> T
% 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/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/path/el__def_c0: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/path/first__def: !p. h4/path/first p = h4/pair/FST (h4/path/fromPath p)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/path/path__cases: !p. (?x. p = h4/path/stopped__at x) \/ (?x r q. p = h4/path/pcons x r q)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/path/take__def__compute_c2: !p n. h4/path/take (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) p = h4/path/pcons (h4/path/first p) (h4/path/first__label p) (h4/path/take (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/path/tail p))
% Assm: h4/path/take__def__compute_c1: !p n. h4/path/take (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) p = h4/path/pcons (h4/path/first p) (h4/path/first__label p) (h4/path/take (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/path/tail p))
% Assm: h4/path/first__thm_c1: !x r p. h4/path/first (h4/path/pcons x r p) = x
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/path/first__take: !p i. h4/path/first (h4/path/take i p) = h4/path/first p
% Assm: h4/path/pconcat__def: !p2 p1 lab. h4/path/pconcat p1 lab p2 = h4/path/toPath (h4/pair/_2C (h4/path/first p1) (h4/llist/LAPPEND (h4/pair/SND (h4/path/fromPath p1)) (h4/llist/LCONS (h4/pair/_2C lab (h4/path/first p2)) (h4/pair/SND (h4/path/fromPath p2)))))
% Assm: h4/path/el__def__compute_c0: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm: h4/path/take__def__compute_c0: !p. h4/path/take h4/num/0 p = h4/path/stopped__at (h4/path/first p)
% Assm: h4/path/firstP__at__zero: !p P. h4/path/firstP__at P p h4/num/0 <=> P (h4/path/first p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/dc__disj: !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__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/path/el__def_c1: !p n. h4/path/el (h4/num/SUC n) p = h4/path/el n (h4/path/tail p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/path/not__every: !p P. ~h4/path/every P p <=> h4/path/exists (h4/combin/o $not P) p
% Assm: h4/bool/DISJ__IMP__THM: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/path/first__thm_c0: !x. h4/path/first (h4/path/stopped__at x) = x
% Assm: h4/path/first__drop: !p i. h4/bool/IN i (h4/path/PL p) ==> h4/path/first (h4/path/drop i p) = h4/path/el i p
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/path/path__rep__bijections__thm_c1: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/path/first__seg: !p j i. h4/arithmetic/_3C_3D i j /\ h4/bool/IN j (h4/path/PL p) ==> h4/path/first (h4/path/seg i j p) = h4/path/el i p
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/path/filter__def_c1: !x r p P. h4/path/filter P (h4/path/pcons x r p) = h4/bool/COND (P x) (h4/bool/COND (h4/path/exists P p) (h4/path/pcons x r (h4/path/filter P p)) (h4/path/stopped__at x)) (h4/path/filter P p)
% Assm: h4/path/firstP__at__thm_c1: !x r p n P. h4/path/firstP__at P (h4/path/pcons x r p) n <=> n = h4/num/0 /\ P x \/ h4/prim__rec/_3C h4/num/0 n /\ ~P x /\ h4/path/firstP__at P p (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% 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/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/arithmetic/SUB__0_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/path/every__def: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Assm: h4/path/exists__def: !p P. h4/path/exists P p <=> (?i. h4/path/firstP__at P p i)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/arithmetic/SUC__SUB1: !m. h4/arithmetic/_2D (h4/num/SUC m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/bool/RIGHT__FORALL__IMP__THM: !Q P. (!x. P ==> Q x) <=> P ==> (!x. Q x)
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~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/path/toPath__onto: !a. ?r. a = h4/path/toPath r
% Assm: h4/llist/LFINITE__STRONG__INDUCTION: !P. P h4/llist/LNIL /\ (!h t. h4/llist/LFINITE t /\ P t ==> P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> P a0)
% Assm: h4/path/finite__def: !sigma. h4/path/finite sigma <=> h4/llist/LFINITE (h4/pair/SND (h4/path/fromPath sigma))
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FORALL__PROD: !P. (!p. P p) <=> (!p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f 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/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% 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/path/drop__def_c1: !p n. h4/path/drop (h4/num/SUC n) p = h4/path/drop n (h4/path/tail p)
% Assm: h4/path/drop__def_c0: !p. h4/path/drop h4/num/0 p = p
% Assm: h4/path/PL__0: !p. h4/bool/IN h4/num/0 (h4/path/PL p)
% Assm: h4/path/PL__thm_c1: !x r q. h4/path/PL (h4/path/pcons x r q) = h4/pred__set/INSERT h4/num/0 (h4/pred__set/IMAGE h4/num/SUC (h4/path/PL q))
% 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/path/tail__def: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/path/seg__def: !p j i. h4/path/seg i j p = h4/path/take (h4/arithmetic/_2D j i) (h4/path/drop i p)
% Assm: h4/path/PL__downward__closed: !p i. h4/bool/IN i (h4/path/PL p) ==> (!j. h4/prim__rec/_3C j i ==> h4/bool/IN j (h4/path/PL p))
% Assm: h4/arithmetic/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% 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/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_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))
% Goal: !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)
%   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_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_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (happ h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_paths_okpathu_u_def]: !R. h4/path/okpath R = h4/fixedPoint/gfp (h4/path/okpath__f R)
% Assm [h4s_paths_everyu_u_coinduction]: !Q P. (!x. happ P (h4/path/stopped__at x) ==> happ Q x) /\ (!x r p. happ P (h4/path/pcons x r p) ==> happ Q x /\ happ P p) ==> (!p. happ P p ==> h4/path/every Q p)
% Assm [h4s_paths_takeu_u_defu_c1]: !p n. h4/path/take (happ h4/num/SUC n) p = h4/path/pcons (h4/path/first p) (h4/path/first__label p) (h4/path/take n (h4/path/tail p))
% 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_paths_takeu_u_defu_c0]: !p. h4/path/take h4/num/0 p = h4/path/stopped__at (h4/path/first p)
% Assm [h4s_paths_EXISTSu_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_paths_finiteu_u_pathu_u_ind]: !P. (!x. happ P (h4/path/stopped__at x)) /\ (!x r p. h4/path/finite p /\ happ P p ==> happ P (h4/path/pcons x r p)) ==> (!q. h4/path/finite q ==> happ P q)
% Assm [h4s_paths_existsu_u_induction]: !Q P. (!x. happ Q x ==> happ P (h4/path/stopped__at x)) /\ (!x r p. happ Q x ==> happ P (h4/path/pcons x r p)) /\ (!x r p. happ P p ==> happ P (h4/path/pcons x r p)) ==> (!p. h4/path/exists Q p ==> happ P p)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS]: !g f. (!n. happ g (happ h4/num/SUC n) = happ (happ f n) (happ h4/num/SUC n)) <=> (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = happ (happ f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = happ (happ f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. 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_REFLu_u_CLAUSE]: !x. x = x <=> T
% 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_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_paths_elu_u_defu_c0]: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_paths_firstu_u_def]: !p. h4/path/first p = h4/pair/FST (h4/path/fromPath p)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_paths_pathu_u_cases]: !p. (?x. p = h4/path/stopped__at x) \/ (?x r q. p = h4/path/pcons x r q)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_paths_takeu_u_defu_u_computeu_c2]: !p n. h4/path/take (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) p = h4/path/pcons (h4/path/first p) (h4/path/first__label p) (h4/path/take (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/path/tail p))
% Assm [h4s_paths_takeu_u_defu_u_computeu_c1]: !p n. h4/path/take (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) p = h4/path/pcons (h4/path/first p) (h4/path/first__label p) (h4/path/take (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/path/tail p))
% Assm [h4s_paths_firstu_u_thmu_c1]: !x r p. h4/path/first (h4/path/pcons x r p) = x
% 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_paths_firstu_u_take]: !p i. h4/path/first (h4/path/take i p) = h4/path/first p
% Assm [h4s_paths_pconcatu_u_def]: !p2 p1 lab. h4/path/pconcat p1 lab p2 = h4/path/toPath (h4/pair/_2C (h4/path/first p1) (h4/llist/LAPPEND (h4/pair/SND (h4/path/fromPath p1)) (h4/llist/LCONS (h4/pair/_2C lab (h4/path/first p2)) (h4/pair/SND (h4/path/fromPath p2)))))
% Assm [h4s_paths_elu_u_defu_u_computeu_c0]: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm [h4s_paths_takeu_u_defu_u_computeu_c0]: !p. h4/path/take h4/num/0 p = h4/path/stopped__at (h4/path/first p)
% Assm [h4s_paths_firstPu_u_atu_u_zero]: !p P. h4/path/firstP__at P p h4/num/0 <=> happ P (h4/path/first p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_dcu_u_disj]: !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_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~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_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_paths_elu_u_defu_c1]: !p n. h4/path/el (happ h4/num/SUC n) p = h4/path/el n (h4/path/tail p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_paths_notu_u_every]: !p P. ~h4/path/every P p <=> h4/path/exists (h4/combin/o $not P) p
% Assm [h4s_bools_DISJu_u_IMPu_u_THM]: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_paths_firstu_u_thmu_c0]: !x. h4/path/first (h4/path/stopped__at x) = x
% Assm [h4s_paths_firstu_u_drop]: !p i. h4/bool/IN i (h4/path/PL p) ==> h4/path/first (h4/path/drop i p) = h4/path/el i p
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_paths_pathu_u_repu_u_bijectionsu_u_thmu_c1]: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_paths_firstu_u_seg]: !p j i. h4/arithmetic/_3C_3D i j /\ h4/bool/IN j (h4/path/PL p) ==> h4/path/first (h4/path/seg i j p) = h4/path/el i p
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% 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_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_paths_filteru_u_defu_c1]: !x r p P. h4/path/filter P (h4/path/pcons x r p) = h4/bool/COND (happ P x) (h4/bool/COND (h4/path/exists P p) (h4/path/pcons x r (h4/path/filter P p)) (h4/path/stopped__at x)) (h4/path/filter P p)
% Assm [h4s_paths_firstPu_u_atu_u_thmu_c1]: !x r p n P. h4/path/firstP__at P (h4/path/pcons x r p) n <=> n = h4/num/0 /\ happ P x \/ h4/prim__rec/_3C h4/num/0 n /\ ~happ P x /\ h4/path/firstP__at P p (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% 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_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_arithmetics_SUBu_u_0u_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = happ h4/num/SUC n)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(happ h4/num/SUC n = h4/num/0)
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (happ h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% 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_existsu_u_def]: !p P. h4/path/exists P p <=> (?i. h4/path/firstP__at P p i)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_arithmetics_SUCu_u_SUB1]: !m. h4/arithmetic/_2D (happ h4/num/SUC m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (happ h4/num/SUC n)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. P ==> happ Q x) <=> P ==> (!x. happ Q x)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~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_paths_toPathu_u_onto]: !a. ?r. a = h4/path/toPath r
% Assm [h4s_llists_LFINITEu_u_STRONGu_u_INDUCTION]: !P. happ P h4/llist/LNIL /\ (!h t. h4/llist/LFINITE t /\ happ P t ==> happ P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> happ P a0)
% Assm [h4s_paths_finiteu_u_def]: !sigma. h4/path/finite sigma <=> h4/llist/LFINITE (h4/pair/SND (h4/path/fromPath sigma))
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_FORALLu_u_PROD]: !P. (!p. happ P p) <=> (!p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% 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_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_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% 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_paths_dropu_u_defu_c1]: !p n. h4/path/drop (happ h4/num/SUC n) p = h4/path/drop n (h4/path/tail p)
% Assm [h4s_paths_dropu_u_defu_c0]: !p. h4/path/drop h4/num/0 p = p
% Assm [h4s_paths_PLu_u_0]: !p. h4/bool/IN h4/num/0 (h4/path/PL p)
% Assm [h4s_paths_PLu_u_thmu_c1]: !x r q. h4/path/PL (h4/path/pcons x r q) = h4/pred__set/INSERT h4/num/0 (h4/pred__set/IMAGE h4/num/SUC (h4/path/PL q))
% 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_paths_tailu_u_def]: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. happ h4/num/SUC m = happ h4/num/SUC n <=> m = n
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_paths_segu_u_def]: !p j i. h4/path/seg i j p = h4/path/take (h4/arithmetic/_2D j i) (h4/path/drop i p)
% Assm [h4s_paths_PLu_u_downwardu_u_closed]: !p i. h4/bool/IN i (h4/path/PL p) ==> (!j. h4/prim__rec/_3C j i ==> h4/bool/IN j (h4/path/PL p))
% 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_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_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_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))
% Goal: !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)
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_Q1230050,TV_Q1230046]: ![V_f, V_g]: (![V_x]: s(TV_Q1230046,happ(s(t_fun(TV_Q1230050,TV_Q1230046),V_f),s(TV_Q1230050,V_x))) = s(TV_Q1230046,happ(s(t_fun(TV_Q1230050,TV_Q1230046),V_g),s(TV_Q1230050,V_x))) => s(t_fun(TV_Q1230050,TV_Q1230046),V_f) = s(t_fun(TV_Q1230050,TV_Q1230046),V_g))).
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_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),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_fn),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
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_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_everyu_u_coinduction, axiom, ![TV_u_27a,TV_u_27b]: ![V_Q, 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)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),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)))))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,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),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,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_Q),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))).
fof(ah4s_paths_takeu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))),s(TV_u_27b,h4s_paths_firstu_u_label(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_take(s(t_h4s_nums_num,V_n),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))).
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_paths_takeu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,h4s_nums_0),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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_EXISTSu_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_paths_finiteu_u_pathu_u_ind, axiom, ![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,h4s_paths_finite(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))))) => 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)))))))) => ![V_q]: (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q)))) => 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_q))))))).
fof(ah4s_paths_existsu_u_induction, axiom, ![TV_u_27a,TV_u_27b]: ![V_Q, 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)))) => 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(TV_u_27a,t_bool),V_Q),s(TV_u_27a,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_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))))))) & ![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),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))))))))) => ![V_p]: (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_Q),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))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS, axiom, ![TV_u_27a]: ![V_g, V_f]: (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),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_nums_suc),s(t_h4s_nums_num,V_n))))) <=> (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(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(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(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(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))))))))).
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_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_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_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_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_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_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_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_firstu_u_def, axiom, ![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))) = s(TV_u_27a,h4s_pairs_fst(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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_paths_pathu_u_cases, axiom, ![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))) | ?[V_x, V_r, V_q]: 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_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))).
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_paths_takeu_u_defu_u_computeu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))),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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))),s(TV_u_27b,h4s_paths_firstu_u_label(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_take(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ah4s_paths_takeu_u_defu_u_computeu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))),s(TV_u_27b,h4s_paths_firstu_u_label(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_take(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),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_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ah4s_paths_firstu_u_thmu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_r, V_p]: s(TV_u_27a,h4s_paths_first(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(TV_u_27a,V_x)).
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_paths_firstu_u_take, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_i]: s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,V_i),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_paths_pconcatu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1, V_lab]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pconcat(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(TV_u_27b,V_lab),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))) = 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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_llists_lappend(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_p1))))),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_lab),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))))),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_p2)))))))))))))).
fof(ah4s_paths_elu_u_defu_u_computeu_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_paths_takeu_u_defu_u_computeu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,h4s_nums_0),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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_firstPu_u_atu_u_zero, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_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,h4s_nums_0))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),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_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_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_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_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_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (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_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_paths_elu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_n),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_paths_notu_u_every, 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_bools_DISJu_u_IMPu_u_THM, 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_R))) & (p(s(t_bool,V_Q)) => p(s(t_bool,V_R)))))).
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_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_firstu_u_drop, axiom, ![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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_drop(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),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))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_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_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_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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(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_27a,V_x)).
fof(ah4s_paths_firstu_u_seg, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_j, V_i]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_j)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_j),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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_seg(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_j),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),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))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_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_paths_filteru_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p, V_P]: 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_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_bools_cond(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_bools_cond(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))),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),h4s_paths_filter(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_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_filter(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_firstPu_u_atu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p, 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_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_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))))) | (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(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) & 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,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),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_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_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_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_SUBu_u_0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
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_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[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_nums_suc),s(t_h4s_nums_num,V_n))))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
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,happ(s(t_fun(t_h4s_nums_num,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_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_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![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))))) <=> (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))))))).
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_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_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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = 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_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
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_arithmetics_SUCu_u_SUB1, axiom, ![V_m]: 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_nums_suc),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_nums_num,V_m)).
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_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![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))))) <=> (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))))))).
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_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_paths_toPathu_u_onto, axiom, ![TV_u_27b,TV_u_27a]: ![V_a]: ?[V_r]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_a) = 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)))).
fof(ah4s_llists_LFINITEu_u_STRONGu_u_INDUCTION, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)))) & ![V_h, V_t]: ((p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_t)))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_t))))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),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)))))))) => ![V_a0]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_a0))))))).
fof(ah4s_paths_finiteu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_sigma]: s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_sigma))) = s(t_bool,h4s_llists_lfinite(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_sigma)))))))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[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))))) <=> (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))))))).
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_pairs_FORALLu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))) <=> ![V_pu_u_1, V_pu_u_2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
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_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_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_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_paths_dropu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_drop(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),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_drop(s(t_h4s_nums_num,V_n),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_dropu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_drop(s(t_h4s_nums_num,h4s_nums_0),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_p)).
fof(ah4s_paths_PLu_u_0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,h4s_nums_0),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))))))).
fof(ah4s_paths_PLu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_q]: 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_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_q))))) = 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_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q)))))))).
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_paths_tailu_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_tail(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),V_p)).
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_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_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),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_nums_suc),s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_paths_segu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_j, V_i]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_seg(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_j),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_take(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_i))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_drop(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_PLu_u_downwardu_u_closed, axiom, ![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)))))) => ![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,h4s_bools_in(s(t_h4s_nums_num,V_j),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))))))))).
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_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_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_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(ch4s_paths_okpathu_u_cou_u_ind, conjecture, ![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))))))).
