%   ORIGINAL: h4/path/okpath__unfold
% 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/unfold__thm2_c1: !x v2 v1 proj f. f x = h4/option/SOME (h4/pair/_2C v1 v2) ==> h4/path/unfold proj f x = h4/path/pcons (proj x) v2 (h4/path/unfold proj f v1)
% Assm: h4/path/unfold__def: !s proj f. h4/path/unfold proj f s = h4/path/toPath (h4/pair/_2C (proj s) (h4/llist/LUNFOLD (\s0. h4/option/OPTION__MAP (h4/pair/UNCURRY (\next__s lbl. h4/pair/_2C next__s (h4/pair/_2C lbl (proj next__s)))) (f s0)) s))
% Assm: h4/path/unfold__thm: !s proj f. h4/path/unfold proj f s = h4/option/option__CASE (f s) (h4/path/stopped__at (proj s)) (\v. h4/pair/pair__CASE v (\s_27 l. h4/path/pcons (proj s) l (h4/path/unfold proj f s_27)))
% Assm: h4/path/labels__unfold: !s proj f. h4/path/labels (h4/path/unfold proj f s) = h4/llist/LUNFOLD f s
% Assm: h4/llist/LMAP__LUNFOLD: !s g f. h4/llist/LMAP f (h4/llist/LUNFOLD g s) = h4/llist/LUNFOLD (\s0. h4/option/OPTION__MAP (h4/pair/UNCURRY (\x y. h4/pair/_2C x (f y))) (g s0)) s
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/path/path__rep__bijections__thm_c1: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/path/toPath__11: !r_27 r. h4/path/toPath r = h4/path/toPath r_27 <=> r = r_27
% Assm: h4/path/first__def: !p. h4/path/first p = h4/pair/FST (h4/path/fromPath p)
% 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__11: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/path/stopped__at__def: !x. h4/path/stopped__at x = h4/path/toPath (h4/pair/_2C x h4/llist/LNIL)
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/llist/llist__Axiom__1: !f. ?g. !x. g x = h4/option/option__CASE (f x) h4/llist/LNIL (\v. h4/pair/pair__CASE v (\a b. h4/llist/LCONS b (g a)))
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% 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/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/path/unfold__thm2_c0: !x proj f. f x = h4/option/NONE ==> h4/path/unfold proj f x = h4/path/stopped__at (proj x)
% 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__thm_c1: !x r p R. h4/path/okpath R (h4/path/pcons x r p) <=> R x r (h4/path/first p) /\ h4/path/okpath R p
% Assm: h4/path/pmap__thm_c1: !x r p g f. h4/path/pmap f g (h4/path/pcons x r p) = h4/path/pcons (f x) (g r) (h4/path/pmap f g 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/path/finite__okpath__ind: !R P. (!x. P (h4/path/stopped__at x)) /\ (!x r p. h4/path/okpath R p /\ h4/path/finite p /\ R x r (h4/path/first p) /\ P p ==> P (h4/path/pcons x r p)) ==> (!sigma. h4/path/okpath R sigma /\ h4/path/finite sigma ==> P sigma)
% Assm: h4/path/okpath__pmap: !p g f R. h4/path/okpath R p /\ (!x r y. R x r y ==> R (f x) (g r) (f y)) ==> h4/path/okpath R (h4/path/pmap f g p)
% Assm: h4/path/SN__finite__paths__EQ: !R. h4/path/SN R <=> (!p. h4/path/okpath R p ==> h4/path/finite p)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/option/OPTION__MAP__EQ__SOME: !y x f. h4/option/OPTION__MAP f x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = f z)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/llist/LMAP0_c1: !t h f. h4/llist/LMAP f (h4/llist/LCONS h t) = h4/llist/LCONS (f h) (h4/llist/LMAP f t)
% Assm: h4/bool/EQ__REFL: !x. x = x
% 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__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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/option/OPTION__MAP__EQ__NONE__both__ways_c0: !x f. h4/option/OPTION__MAP f x = h4/option/NONE <=> x = h4/option/NONE
% Assm: h4/path/labels__LMAP: !p. h4/path/labels p = h4/llist/LMAP h4/pair/FST (h4/pair/SND (h4/path/fromPath p))
% Assm: h4/option/OPTION__MAP__COMPOSE: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/llist/LUNFOLD__EQ: !s ll f R. R s ll /\ (!s0 ll0. R s0 ll0 ==> f s0 = h4/option/NONE /\ ll0 = h4/llist/LNIL \/ (?s_27 x ll_27. f s0 = h4/option/SOME (h4/pair/_2C s_27 x) /\ h4/llist/LHD ll0 = h4/option/SOME x /\ h4/llist/LTL ll0 = h4/option/SOME ll_27 /\ R s_27 ll_27)) ==> h4/llist/LUNFOLD f s = ll
% Assm: h4/llist/LMAP0_c0: !f. h4/llist/LMAP f h4/llist/LNIL = h4/llist/LNIL
% Assm: h4/llist/LUNFOLD__THM_c1: !x v2 v1 f. f x = h4/option/SOME (h4/pair/_2C v1 v2) ==> h4/llist/LUNFOLD f x = h4/llist/LCONS v2 (h4/llist/LUNFOLD f v1)
% Assm: h4/llist/LTL__THM_c1: !t h. h4/llist/LTL (h4/llist/LCONS h t) = h4/option/SOME t
% Assm: h4/llist/LHD__THM_c1: !t h. h4/llist/LHD (h4/llist/LCONS h t) = h4/option/SOME h
% Assm: h4/llist/LHD__THM_c0: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/LTL__THM_c0: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm: h4/option/OPTION__MAP__EQ__NONE__both__ways_c1: !x f. h4/option/NONE = h4/option/OPTION__MAP f x <=> x = h4/option/NONE
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/llist/LUNFOLD__THM_c0: !x f. f x = h4/option/NONE ==> h4/llist/LUNFOLD f x = h4/llist/LNIL
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (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/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/path/okpath__monotone: !R. h4/fixedPoint/monotone (h4/path/okpath__f R)
% Assm: h4/path/okpath__def: !R. h4/path/okpath R = h4/fixedPoint/gfp (h4/path/okpath__f R)
% 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/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% 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/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = 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/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/path/pmap__def: !p g f. h4/path/pmap f g p = h4/path/toPath (h4/pair/_23_23 f (h4/llist/LMAP (h4/pair/_23_23 g f)) (h4/path/fromPath p))
% Assm: h4/pair/SND__PAIR__MAP: !p g f. h4/pair/SND (h4/pair/_23_23 f g p) = g (h4/pair/SND p)
% Assm: h4/pair/FST__PAIR__MAP: !p g f. h4/pair/FST (h4/pair/_23_23 f g p) = f (h4/pair/FST p)
% Assm: h4/pair/PAIR__MAP__THM: !y x g f. h4/pair/_23_23 f g (h4/pair/_2C x y) = h4/pair/_2C (f x) (g y)
% Assm: h4/path/stopped__at__11: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/RIGHT__EXISTS__AND__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/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% 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/path/okpath__thm_c0: !x R. h4/path/okpath R (h4/path/stopped__at x)
% Assm: h4/path/SN__finite__paths: !p R. h4/path/SN R /\ h4/path/okpath R p ==> h4/path/finite p
% Assm: h4/path/finite__paths__SN: !R. (!p. h4/path/okpath R p ==> h4/path/finite p) ==> h4/path/SN R
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% 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/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% 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/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/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/path/first__pmap: !p g f. h4/path/first (h4/path/pmap f g p) = f (h4/path/first p)
% Assm: h4/path/pmap__thm_c0: !x g f. h4/path/pmap f g (h4/path/stopped__at x) = h4/path/stopped__at (f x)
% 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/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Goal: !s proj m f P. P s /\ (!s0 s_27 l. P s0 /\ f s0 = h4/option/SOME (h4/pair/_2C s_27 l) ==> P s_27) /\ (!s0 s_27 l. P s0 /\ f s0 = h4/option/SOME (h4/pair/_2C s_27 l) ==> m (proj s0) l (proj s_27)) ==> h4/path/okpath m (h4/path/unfold proj f s)
%   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_unfoldu_u_thm2u_c1]: !x v2 v1 proj f. happ f x = h4/option/SOME (h4/pair/_2C v1 v2) ==> h4/path/unfold proj f x = h4/path/pcons (happ proj x) v2 (h4/path/unfold proj f v1)
% Assm [h4s_paths_unfoldu_u_def]: !_2. (!proj next__s lbl. happ (happ (happ _2 proj) next__s) lbl = h4/pair/_2C next__s (h4/pair/_2C lbl (happ proj next__s))) ==> (!_1. (!proj next__s. happ (happ _1 proj) next__s = happ (happ _2 proj) next__s) ==> (!_0. (!proj f s0. happ (happ (happ _0 proj) f) s0 = h4/option/OPTION__MAP (h4/pair/UNCURRY (happ _1 proj)) (happ f s0)) ==> (!s proj f. h4/path/unfold proj f s = h4/path/toPath (h4/pair/_2C (happ proj s) (h4/llist/LUNFOLD (happ (happ _0 proj) f) s)))))
% Assm [h4s_paths_unfoldu_u_thm]: !_2. (!s proj f s_27 l. happ (happ (happ (happ (happ _2 s) proj) f) s_27) l = h4/path/pcons (happ proj s) l (h4/path/unfold proj f s_27)) ==> (!_1. (!s proj f s_27. happ (happ (happ (happ _1 s) proj) f) s_27 = happ (happ (happ (happ _2 s) proj) f) s_27) ==> (!_0. (!s proj f v. happ (happ (happ (happ _0 s) proj) f) v = h4/pair/pair__CASE v (happ (happ (happ _1 s) proj) f)) ==> (!s proj f. h4/path/unfold proj f s = h4/option/option__CASE (happ f s) (h4/path/stopped__at (happ proj s)) (happ (happ (happ _0 s) proj) f))))
% Assm [h4s_paths_labelsu_u_unfold]: !s proj f. h4/path/labels (h4/path/unfold proj f s) = h4/llist/LUNFOLD f s
% Assm [h4s_llists_LMAPu_u_LUNFOLD]: !_2. (!x f y. happ (happ (happ _2 x) f) y = h4/pair/_2C x (happ f y)) ==> (!_1. (!f x. happ (happ _1 f) x = happ (happ _2 x) f) ==> (!_0. (!f g s0. happ (happ (happ _0 f) g) s0 = h4/option/OPTION__MAP (h4/pair/UNCURRY (happ _1 f)) (happ g s0)) ==> (!s g f. happ (h4/llist/LMAP f) (h4/llist/LUNFOLD g s) = h4/llist/LUNFOLD (happ (happ _0 f) g) s)))
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% 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_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_paths_pathu_u_repu_u_bijectionsu_u_thmu_c1]: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_paths_toPathu_u_11]: !r_27 r. h4/path/toPath r = h4/path/toPath r_27 <=> r = r_27
% Assm [h4s_paths_firstu_u_def]: !p. h4/path/first p = happ h4/pair/FST (h4/path/fromPath p)
% 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_11]: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% 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_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_combins_Cu_u_DEF]: !x x x. h4/combin/C x x x = happ (happ x x) x
% Assm [h4s_llists_llistu_u_Axiomu_u_1]: !_2. (!g a b. happ (happ (happ _2 g) a) b = h4/llist/LCONS b (happ g a)) ==> (!_1. (!g a. happ (happ _1 g) a = happ (happ _2 g) a) ==> (!_0. (!g v. happ (happ _0 g) v = h4/pair/pair__CASE v (happ _1 g)) ==> (!f. ?g. !x. happ g x = h4/option/option__CASE (happ f x) h4/llist/LNIL (happ _0 g))))
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% 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_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_paths_unfoldu_u_thm2u_c0]: !x proj f. happ f x = h4/option/NONE ==> h4/path/unfold proj f x = h4/path/stopped__at (happ proj x)
% 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_thmu_c1]: !x r p R. happ (h4/path/okpath R) (h4/path/pcons x r p) <=> happ (happ (happ R x) r) (h4/path/first p) /\ happ (h4/path/okpath R) p
% Assm [h4s_paths_pmapu_u_thmu_c1]: !x r p g f. h4/path/pmap f g (h4/path/pcons x r p) = h4/path/pcons (happ f x) (happ g r) (h4/path/pmap f g 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_paths_finiteu_u_okpathu_u_ind]: !R P. (!x. happ P (h4/path/stopped__at x)) /\ (!x r p. happ (h4/path/okpath R) p /\ h4/path/finite p /\ happ (happ (happ R x) r) (h4/path/first p) /\ happ P p ==> happ P (h4/path/pcons x r p)) ==> (!sigma. happ (h4/path/okpath R) sigma /\ h4/path/finite sigma ==> happ P sigma)
% Assm [h4s_paths_okpathu_u_pmap]: !p g f R. happ (h4/path/okpath R) p /\ (!x r y. happ (happ (happ R x) r) y ==> happ (happ (happ R (happ f x)) (happ g r)) (happ f y)) ==> happ (h4/path/okpath R) (h4/path/pmap f g p)
% Assm [h4s_paths_SNu_u_finiteu_u_pathsu_u_EQ]: !R. h4/path/SN R <=> (!p. happ (h4/path/okpath R) p ==> h4/path/finite p)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME]: !y x f. h4/option/OPTION__MAP f x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = happ f z)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_llists_LMAP0u_c1]: !t h f. happ (h4/llist/LMAP f) (h4/llist/LCONS h t) = h4/llist/LCONS (happ f h) (happ (h4/llist/LMAP f) t)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% 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_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c0]: !x f. h4/option/OPTION__MAP f x = h4/option/NONE <=> x = h4/option/NONE
% Assm [h4s_paths_labelsu_u_LMAP]: !p. h4/path/labels p = happ (h4/llist/LMAP h4/pair/FST) (h4/pair/SND (h4/path/fromPath p))
% Assm [h4s_options_OPTIONu_u_MAPu_u_COMPOSE]: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_llists_LUNFOLDu_u_EQ]: !s ll f R. happ (happ R s) ll /\ (!s0 ll0. happ (happ R s0) ll0 ==> happ f s0 = h4/option/NONE /\ ll0 = h4/llist/LNIL \/ (?s_27 x ll_27. happ f s0 = h4/option/SOME (h4/pair/_2C s_27 x) /\ h4/llist/LHD ll0 = h4/option/SOME x /\ h4/llist/LTL ll0 = h4/option/SOME ll_27 /\ happ (happ R s_27) ll_27)) ==> h4/llist/LUNFOLD f s = ll
% Assm [h4s_llists_LMAP0u_c0]: !f. happ (h4/llist/LMAP f) h4/llist/LNIL = h4/llist/LNIL
% Assm [h4s_llists_LUNFOLDu_u_THMu_c1]: !x v2 v1 f. happ f x = h4/option/SOME (h4/pair/_2C v1 v2) ==> h4/llist/LUNFOLD f x = h4/llist/LCONS v2 (h4/llist/LUNFOLD f v1)
% Assm [h4s_llists_LTLu_u_THMu_c1]: !t h. h4/llist/LTL (h4/llist/LCONS h t) = h4/option/SOME t
% Assm [h4s_llists_LHDu_u_THMu_c1]: !t h. h4/llist/LHD (h4/llist/LCONS h t) = h4/option/SOME h
% Assm [h4s_llists_LHDu_u_THMu_c0]: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_LTLu_u_THMu_c0]: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c1]: !x f. h4/option/NONE = h4/option/OPTION__MAP f x <=> x = h4/option/NONE
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_llists_LUNFOLDu_u_THMu_c0]: !x f. happ f x = h4/option/NONE ==> h4/llist/LUNFOLD f x = h4/llist/LNIL
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (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_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_paths_okpathu_u_monotone]: !R. h4/fixedPoint/monotone (h4/path/okpath__f R)
% Assm [h4s_paths_okpathu_u_def]: !R. h4/path/okpath R = h4/fixedPoint/gfp (h4/path/okpath__f R)
% 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_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% 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_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_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (happ h4/pair/FST x) (h4/pair/SND x) = 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_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_paths_pmapu_u_def]: !p g f. h4/path/pmap f g p = h4/path/toPath (happ (h4/pair/_23_23 f (h4/llist/LMAP (h4/pair/_23_23 g f))) (h4/path/fromPath p))
% Assm [h4s_pairs_SNDu_u_PAIRu_u_MAP]: !p g f. h4/pair/SND (happ (h4/pair/_23_23 f g) p) = happ g (h4/pair/SND p)
% Assm [h4s_pairs_FSTu_u_PAIRu_u_MAP]: !p g f. happ h4/pair/FST (happ (h4/pair/_23_23 f g) p) = happ f (happ h4/pair/FST p)
% Assm [h4s_pairs_PAIRu_u_MAPu_u_THM]: !y x g f. happ (h4/pair/_23_23 f g) (h4/pair/_2C x y) = h4/pair/_2C (happ f x) (happ g y)
% Assm [h4s_paths_stoppedu_u_atu_u_11]: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_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_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% 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_paths_okpathu_u_thmu_c0]: !x R. happ (h4/path/okpath R) (h4/path/stopped__at x)
% Assm [h4s_paths_SNu_u_finiteu_u_paths]: !p R. h4/path/SN R /\ happ (h4/path/okpath R) p ==> h4/path/finite p
% Assm [h4s_paths_finiteu_u_pathsu_u_SN]: !R. (!p. happ (h4/path/okpath R) p ==> h4/path/finite p) ==> h4/path/SN R
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% 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_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% 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_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_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_paths_firstu_u_pmap]: !p g f. h4/path/first (h4/path/pmap f g p) = happ f (h4/path/first p)
% Assm [h4s_paths_pmapu_u_thmu_c0]: !x g f. h4/path/pmap f g (h4/path/stopped__at x) = h4/path/stopped__at (happ f x)
% 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_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Goal: !s proj m f P. happ P s /\ (!s0 s_27 l. happ P s0 /\ happ f s0 = h4/option/SOME (h4/pair/_2C s_27 l) ==> happ P s_27) /\ (!s0 s_27 l. happ P s0 /\ happ f s0 = h4/option/SOME (h4/pair/_2C s_27 l) ==> happ (happ (happ m (happ proj s0)) l) (happ proj s_27)) ==> happ (h4/path/okpath m) (h4/path/unfold proj f s)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1230641,TV_Q1230637]: ![V_f, V_g]: (![V_x]: s(TV_Q1230637,happ(s(t_fun(TV_Q1230641,TV_Q1230637),V_f),s(TV_Q1230641,V_x))) = s(TV_Q1230637,happ(s(t_fun(TV_Q1230641,TV_Q1230637),V_g),s(TV_Q1230641,V_x))) => s(t_fun(TV_Q1230641,TV_Q1230637),V_f) = s(t_fun(TV_Q1230641,TV_Q1230637),V_g))).
fof(ah4s_paths_unfoldu_u_thm2u_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_v2, V_v1, V_proj, V_f]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_v1),s(TV_u_27c,V_v2))))) => s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_unfold(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pcons(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(TV_u_27a,V_x))),s(TV_u_27c,V_v2),s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_unfold(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_v1))))))).
fof(ah4s_paths_unfoldu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_2]: (![V_proj, V_nextu_u_s, V_lbl]: s(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),V_uu_2),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(TV_u_27c,V_nextu_u_s))),s(TV_u_27b,V_lbl))) = s(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27c,V_nextu_u_s),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_lbl),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_proj),s(TV_u_27c,V_nextu_u_s))))))) => ![V_uu_1]: (![V_proj, V_nextu_u_s]: s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),V_uu_1),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(TV_u_27c,V_nextu_u_s))) = s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),V_uu_2),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(TV_u_27c,V_nextu_u_s))) => ![V_uu_0]: (![V_proj, V_f, V_s0]: s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f))),s(TV_u_27c,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_pairs_uncurry(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),V_uu_1),s(t_fun(TV_u_27c,TV_u_27a),V_proj))))),s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f),s(TV_u_27c,V_s0))))) => ![V_s, V_proj, V_f]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_unfold(s(t_fun(TV_u_27c,TV_u_27a),V_proj),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f),s(TV_u_27c,V_s))) = 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,happ(s(t_fun(TV_u_27c,TV_u_27a),V_proj),s(TV_u_27c,V_s))),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_llists_lunfold(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f))),s(TV_u_27c,V_s))))))))))).
fof(ah4s_paths_unfoldu_u_thm, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_2]: (![V_s, V_proj, V_f, V_su_27, V_l]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)))))),V_uu_2),s(TV_u_27a,V_s))),s(t_fun(TV_u_27a,TV_u_27b),V_proj))),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f))),s(TV_u_27a,V_su_27))),s(TV_u_27c,V_l))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pcons(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(TV_u_27a,V_s))),s(TV_u_27c,V_l),s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_unfold(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_su_27))))) => ![V_uu_1]: (![V_s, V_proj, V_f, V_su_27]: s(t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)))))),V_uu_1),s(TV_u_27a,V_s))),s(t_fun(TV_u_27a,TV_u_27b),V_proj))),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f))),s(TV_u_27a,V_su_27))) = s(t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)))))),V_uu_2),s(TV_u_27a,V_s))),s(t_fun(TV_u_27a,TV_u_27b),V_proj))),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f))),s(TV_u_27a,V_su_27))) => ![V_uu_0]: (![V_s, V_proj, V_f, V_v]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_paths_path(TV_u_27b,TV_u_27c))))),V_uu_0),s(TV_u_27a,V_s))),s(t_fun(TV_u_27a,TV_u_27b),V_proj))),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),V_v))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),V_v),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27c,t_h4s_paths_path(TV_u_27b,TV_u_27c)))))),V_uu_1),s(TV_u_27a,V_s))),s(t_fun(TV_u_27a,TV_u_27b),V_proj))),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f))))) => ![V_s, V_proj, V_f]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_unfold(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s))),s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_stoppedu_u_at(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(TV_u_27a,V_s))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_paths_path(TV_u_27b,TV_u_27c))))),V_uu_0),s(TV_u_27a,V_s))),s(t_fun(TV_u_27a,TV_u_27b),V_proj))),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f))))))))).
fof(ah4s_paths_labelsu_u_unfold, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_s, V_proj, V_f]: s(t_h4s_llists_llist(TV_u_27c),h4s_paths_labels(s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_unfold(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s))))) = s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s)))).
fof(ah4s_llists_LMAPu_u_LUNFOLD, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_2]: (![V_x, V_f, V_y]: s(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_2),s(TV_u_27c,V_x))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),h4s_pairs_u_2c(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,V_y))))) => ![V_uu_1]: (![V_f, V_x]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27c,V_x))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_2),s(TV_u_27c,V_x))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) => ![V_uu_0]: (![V_f, V_g, V_s0]: s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_g))),s(TV_u_27c,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),h4s_pairs_uncurry(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))))),s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),happ(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_g),s(TV_u_27c,V_s0))))) => ![V_s, V_g, V_f]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_llists_llist(TV_u_27b)),h4s_llists_lmap(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lunfold(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_g),s(TV_u_27c,V_s))))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_g))),s(TV_u_27c,V_s))))))).
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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_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_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_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(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,V_q),s(TV_u_27b,V_r)))).
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_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
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,f0))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (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_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_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_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_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
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_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_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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),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_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_11, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_h2, V_h1]: (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h1),s(t_h4s_llists_llist(TV_u_27a),V_t1))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h2),s(t_h4s_llists_llist(TV_u_27a),V_t2))) <=> (s(TV_u_27a,V_h1) = s(TV_u_27a,V_h2) & s(t_h4s_llists_llist(TV_u_27a),V_t1) = s(t_h4s_llists_llist(TV_u_27a),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_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_bools_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t)).
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_options_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(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_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27b,V_x0),s(TV_u_27a,V_x1))) = 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_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_llists_llistu_u_Axiomu_u_1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_g, V_a, V_b]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))),s(TV_u_27a,V_a))),s(TV_u_27b,V_b))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_b),s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(TV_u_27a,V_a))))) => ![V_uu_1]: (![V_g, V_a]: s(t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))),s(TV_u_27a,V_a))) = s(t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))),s(TV_u_27a,V_a))) => ![V_uu_0]: (![V_g, V_v]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))) = s(t_h4s_llists_llist(TV_u_27b),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))))) => ![V_f]: ?[V_g]: ![V_x]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(TV_u_27a,V_x))) = s(t_h4s_llists_llist(TV_u_27b),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))),s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))))))))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
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_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_paths_unfoldu_u_thm2u_c0, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_proj, V_f]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_options_none) => s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_unfold(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_stoppedu_u_at(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(TV_u_27a,V_x))))))).
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_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p, 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_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),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_pmapu_u_thmu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a,TV_u_27d]: ![V_x, V_r, V_p, V_g, V_f]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27d,TV_u_27c),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27d,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),V_p))))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pcons(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_27c,happ(s(t_fun(TV_u_27d,TV_u_27c),V_g),s(TV_u_27d,V_r))),s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27d,TV_u_27c),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),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_paths_finiteu_u_okpathu_u_ind, axiom, ![TV_u_27a,TV_u_27b]: ![V_R, 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),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)))) & (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(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))))))) => 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_sigma]: ((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_sigma)))) & p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_sigma))))) => 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_sigma))))))).
fof(ah4s_paths_okpathu_u_pmap, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f, 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_p)))) & ![V_x, V_r, V_y]: (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,V_y)))) => 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,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),V_g),s(TV_u_27b,V_r))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y)))))))) => 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_pmap(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_fun(TV_u_27b,TV_u_27b),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ah4s_paths_SNu_u_finiteu_u_pathsu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_R]: (p(s(t_bool,h4s_paths_sn(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R)))) <=> ![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)))) => p(s(t_bool,h4s_paths_finite(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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f]: (s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_y))) <=> ?[V_z]: (s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_z))) & 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_z)))))).
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_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_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_llists_LMAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_llists_llist(TV_u_27b)),h4s_llists_lmap(s(t_fun(TV_u_27a,TV_u_27b),V_f))),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_27b),h4s_llists_lcons(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))),s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_llists_llist(TV_u_27b)),h4s_llists_lmap(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_llists_llist(TV_u_27a),V_t)))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_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_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_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_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> s(t_h4s_options_option(TV_u_27b),V_x) = s(t_h4s_options_option(TV_u_27b),h4s_options_none))).
fof(ah4s_paths_labelsu_u_LMAP, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(t_h4s_llists_llist(TV_u_27b),h4s_paths_labels(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_llists_llist(TV_u_27b)),h4s_llists_lmap(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27b),h4s_pairs_fst))),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_options_OPTIONu_u_MAPu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(t_h4s_options_option(TV_u_27a),V_x)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f0)))).
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_llists_LUNFOLDu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_ll, V_f, V_R]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_llists_llist(TV_u_27b),t_bool)),V_R),s(TV_u_27a,V_s))),s(t_h4s_llists_llist(TV_u_27b),V_ll)))) & ![V_s0, V_ll0]: (p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_llists_llist(TV_u_27b),t_bool)),V_R),s(TV_u_27a,V_s0))),s(t_h4s_llists_llist(TV_u_27b),V_ll0)))) => ((s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_options_none) & s(t_h4s_llists_llist(TV_u_27b),V_ll0) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil)) | ?[V_su_27, V_x, V_llu_27]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_su_27),s(TV_u_27b,V_x))))) & (s(t_h4s_options_option(TV_u_27b),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27b),V_ll0))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_x))) & (s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27b),V_ll0))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_options_some(s(t_h4s_llists_llist(TV_u_27b),V_llu_27))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_llists_llist(TV_u_27b),t_bool)),V_R),s(TV_u_27a,V_su_27))),s(t_h4s_llists_llist(TV_u_27b),V_llu_27)))))))))) => s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_s))) = s(t_h4s_llists_llist(TV_u_27b),V_ll))).
fof(ah4s_llists_LMAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_llists_llist(TV_u_27b)),h4s_llists_lmap(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil)).
fof(ah4s_llists_LUNFOLDu_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v2, V_v1, V_f]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_v1),s(TV_u_27b,V_v2))))) => s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_v2),s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_v1))))))).
fof(ah4s_llists_LTLu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_h),s(t_h4s_llists_llist(TV_u_27b),V_t))))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_options_some(s(t_h4s_llists_llist(TV_u_27b),V_t)))).
fof(ah4s_llists_LHDu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(TV_u_27b),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_h),s(t_h4s_llists_llist(TV_u_27b),V_t))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_h)))).
fof(ah4s_llists_LHDu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_llists_LTLu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(t_h4s_options_option(TV_u_27b),V_x))) <=> s(t_h4s_options_option(TV_u_27b),V_x) = s(t_h4s_options_option(TV_u_27b),h4s_options_none))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_llists_LUNFOLDu_u_THMu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_options_none) => s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
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,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
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_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_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_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_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_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_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_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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_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_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_paths_pmapu_u_def, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27c,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27d,TV_u_27c))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_h4s_pairs_prod(TV_u_27c,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27d,TV_u_27c)))),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27d,TV_u_27c))),h4s_llists_lmap(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_h4s_pairs_prod(TV_u_27d,TV_u_27c)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27a,TV_u_27c),V_f))))))),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_pairs_SNDu_u_PAIRu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27d,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_pairs_FSTu_u_PAIRu_u_MAP, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),TV_u_27c),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_pairs_PAIRu_u_MAPu_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_y, V_x, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27c,V_y))))) = s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_2c(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_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_g),s(TV_u_27c,V_y)))))).
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_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_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,f0))))).
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_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_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_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_paths_okpathu_u_thmu_c0, axiom, ![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))))))).
fof(ah4s_paths_SNu_u_finiteu_u_paths, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_R]: ((p(s(t_bool,h4s_paths_sn(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),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_p))))) => p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_finiteu_u_pathsu_u_SN, axiom, ![TV_u_27b,TV_u_27a]: ![V_R]: (![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)))) => p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) => p(s(t_bool,h4s_paths_sn(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R)))))).
fof(ah4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> ?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_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_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_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_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_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_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_paths_firstu_u_pmap, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27c,h4s_paths_first(s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_pmapu_u_thmu_c0, axiom, ![TV_u_27d,TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27d,TV_u_27c),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_stoppedu_u_at(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
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_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(ch4s_paths_okpathu_u_unfold, conjecture, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_s, V_proj, V_m, V_f, V_P]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s)))) & (![V_s0, V_su_27, V_l]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s0)))) & s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_su_27),s(TV_u_27c,V_l)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_su_27))))) & ![V_s0, V_su_27, V_l]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s0)))) & s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c)),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_su_27),s(TV_u_27c,V_l)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool))),V_m),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(TV_u_27a,V_s0))))),s(TV_u_27c,V_l))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(TV_u_27a,V_su_27))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27b,TV_u_27c),t_bool),h4s_paths_okpath(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool))),V_m))),s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_unfold(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s)))))))).
