%   ORIGINAL: h4/path/labels__plink
% 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/labels__def_c1: !x r p. h4/path/labels (h4/path/pcons x r p) = h4/llist/LCONS r (h4/path/labels p)
% 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/path/plink__def_c1: !x r p2 p1. h4/path/plink (h4/path/pcons x r p1) p2 = h4/path/pcons x r (h4/path/plink p1 p2)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/path/first__plink: !p2 p1. h4/path/last p1 = h4/path/first p2 ==> h4/path/first (h4/path/plink p1 p2) = h4/path/first p1
% Assm: h4/path/last__plink: !p2 p1. h4/path/finite p1 /\ h4/path/finite p2 /\ h4/path/last p1 = h4/path/first p2 ==> h4/path/last (h4/path/plink p1 p2) = h4/path/last p2
% Assm: h4/path/okpath__plink: !p2 p1 R. h4/path/finite p1 /\ h4/path/last p1 = h4/path/first p2 ==> (h4/path/okpath R (h4/path/plink p1 p2) <=> h4/path/okpath R p1 /\ h4/path/okpath R p2)
% Assm: h4/path/finite__plink: !p2 p1. h4/path/finite (h4/path/plink p1 p2) <=> h4/path/finite p1 /\ h4/path/finite p2
% Assm: h4/path/labels__def_c0: !x. h4/path/labels (h4/path/stopped__at x) = h4/llist/LNIL
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/path/plink__def_c0: !x p. h4/path/plink (h4/path/stopped__at x) p = p
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% 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/llist/LCONS__11: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm: h4/path/stopped__at__not__pcons_c0: !y x r p. ~(h4/path/stopped__at x = h4/path/pcons y r p)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/path/tail__def: !x r p. h4/path/tail (h4/path/pcons x r p) = 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/path/stopped__at__11: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/path/first__label__def: !x r p. h4/path/first__label (h4/path/pcons x r p) = r
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~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/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% 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/path/last__thm_c0: !x. h4/path/last (h4/path/stopped__at x) = x
% Assm: h4/path/last__thm_c1: !x r p. h4/path/last (h4/path/pcons x r p) = h4/path/last p
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/path/first__thm_c1: !x r p. h4/path/first (h4/path/pcons x r p) = x
% Assm: h4/path/first__thm_c0: !x. h4/path/first (h4/path/stopped__at x) = 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/path/finite__path__ind: !P. (!x. P (h4/path/stopped__at x)) /\ (!x r p. h4/path/finite p /\ P p ==> P (h4/path/pcons x r p)) ==> (!q. h4/path/finite q ==> P q)
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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/path/pconcat__def: !p2 p1 lab. h4/path/pconcat p1 lab p2 = h4/path/toPath (h4/pair/_2C (h4/path/first p1) (h4/llist/LAPPEND (h4/pair/SND (h4/path/fromPath p1)) (h4/llist/LCONS (h4/pair/_2C lab (h4/path/first p2)) (h4/pair/SND (h4/path/fromPath p2)))))
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/llist/LAPPEND__EQ__LNIL: !l2 l1. h4/llist/LAPPEND l1 l2 = h4/llist/LNIL <=> l1 = h4/llist/LNIL /\ l2 = h4/llist/LNIL
% Assm: h4/llist/LCONS__NOT__NIL_c0: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm: h4/llist/LFLATTEN__APPEND: !t h. h4/llist/LFLATTEN (h4/llist/LCONS h t) = h4/llist/LAPPEND h (h4/llist/LFLATTEN t)
% Assm: h4/llist/LTAKE__DROP_c0: !n ll. ~h4/llist/LFINITE ll ==> h4/llist/LAPPEND (h4/llist/fromList (h4/option/THE (h4/llist/LTAKE n ll))) (h4/option/THE (h4/llist/LDROP n ll)) = ll
% 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/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/EQ__REFL: !x. x = x
% 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/path/stopped__at__def: !x. h4/path/stopped__at x = h4/path/toPath (h4/pair/_2C x h4/llist/LNIL)
% Assm: h4/llist/LLIST__EQ: !g f. (!x. f x = h4/llist/LNIL /\ g x = h4/llist/LNIL \/ (?h y. f x = h4/llist/LCONS h (f y) /\ g x = h4/llist/LCONS h (g y))) ==> (!x. f x = g x)
% Assm: h4/path/path__rep__bijections__thm_c1: !r. h4/path/fromPath (h4/path/toPath r) = r
% 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/llist/LMAP0_c0: !f. h4/llist/LMAP f h4/llist/LNIL = h4/llist/LNIL
% 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/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/path/is__stopped__thm_c1: !x r p. h4/path/is__stopped (h4/path/pcons x r p) <=> F
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/path/is__stopped__thm_c0: !x. h4/path/is__stopped (h4/path/stopped__at x) <=> T
% Assm: h4/path/path__Axiom: !f. ?g. !x. g x = h4/pair/pair__CASE (f x) (\y v2. h4/option/option__CASE v2 (h4/path/stopped__at y) (\v3. h4/pair/pair__CASE v3 (\l v. h4/path/pcons y l (g v))))
% Assm: h4/path/path__bisimulation: !p2 p1. p1 = p2 <=> (?R. R p1 p2 /\ (!q1 q2. R q1 q2 ==> (?x. q1 = h4/path/stopped__at x /\ q2 = h4/path/stopped__at x) \/ (?x r q1_27 q2_27. q1 = h4/path/pcons x r q1_27 /\ q2 = h4/path/pcons x r q2_27 /\ R q1_27 q2_27)))
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/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/okpath__thm_c0: !x R. h4/path/okpath R (h4/path/stopped__at x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/path/finite__thm_c1: !x r p. h4/path/finite (h4/path/pcons x r p) <=> h4/path/finite p
% Assm: h4/path/finite__thm_c0: !x. h4/path/finite (h4/path/stopped__at x) <=> T
% Assm: h4/llist/LAPPEND0_c0: !x. h4/llist/LAPPEND h4/llist/LNIL x = x
% Assm: h4/llist/LAPPEND0_c1: !x t h. h4/llist/LAPPEND (h4/llist/LCONS h t) x = h4/llist/LCONS h (h4/llist/LAPPEND t x)
% Assm: h4/llist/llist__CASES: !l. l = h4/llist/LNIL \/ (?h t. l = h4/llist/LCONS h t)
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/llist/LFLATTEN__THM_c2: !tl t h. h4/llist/LFLATTEN (h4/llist/LCONS (h4/llist/LCONS h t) tl) = h4/llist/LCONS h (h4/llist/LFLATTEN (h4/llist/LCONS t tl))
% Assm: h4/llist/LFLATTEN__THM_c1: !t. h4/llist/LFLATTEN (h4/llist/LCONS h4/llist/LNIL t) = h4/llist/LFLATTEN t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/llist/LLIST__STRONG__BISIMULATION: !ll2 ll1. ll1 = ll2 <=> (?R. R ll1 ll2 /\ (!ll3 ll4. R ll3 ll4 ==> ll3 = ll4 \/ (?h t1 t2. ll3 = h4/llist/LCONS h t1 /\ ll4 = h4/llist/LCONS h t2 /\ R t1 t2)))
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/llist/NOT__LFINITE__TAKE: !ll. ~h4/llist/LFINITE ll ==> (!n. ?y. h4/llist/LTAKE n ll = h4/option/SOME y)
% Assm: h4/llist/fromList0_c0: h4/llist/fromList h4/list/NIL = h4/llist/LNIL
% Assm: h4/llist/fromList0_c1: !t h. h4/llist/fromList (h4/list/CONS h t) = h4/llist/LCONS h (h4/llist/fromList t)
% Assm: h4/llist/LFINITE__THM_c1: !t h. h4/llist/LFINITE (h4/llist/LCONS h t) <=> h4/llist/LFINITE t
% Assm: h4/llist/LDROP__THM_c0: !ll. h4/llist/LDROP h4/num/0 ll = h4/option/SOME ll
% Assm: h4/llist/LFINITE__THM_c0: h4/llist/LFINITE h4/llist/LNIL <=> T
% Assm: h4/llist/LDROP__THM_c2: !t n h. h4/llist/LDROP (h4/num/SUC n) (h4/llist/LCONS h t) = h4/llist/LDROP n t
% Assm: h4/llist/LTAKE__THM_c2: !t n h. h4/llist/LTAKE (h4/num/SUC n) (h4/llist/LCONS h t) = h4/option/OPTION__MAP (h4/list/CONS h) (h4/llist/LTAKE n t)
% Assm: h4/llist/LTAKE__THM_c0: !l. h4/llist/LTAKE h4/num/0 l = h4/option/SOME h4/list/NIL
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% 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/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Goal: !p2 p1. h4/path/labels (h4/path/plink p1 p2) = h4/llist/LAPPEND (h4/path/labels p1) (h4/path/labels p2)
%   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_labelsu_u_defu_c1]: !x r p. h4/path/labels (h4/path/pcons x r p) = h4/llist/LCONS r (h4/path/labels p)
% Assm [h4s_paths_labelsu_u_LMAP]: !p. h4/path/labels p = h4/llist/LMAP h4/pair/FST (h4/pair/SND (h4/path/fromPath p))
% Assm [h4s_paths_plinku_u_defu_c1]: !x r p2 p1. h4/path/plink (h4/path/pcons x r p1) p2 = h4/path/pcons x r (h4/path/plink p1 p2)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_paths_firstu_u_plink]: !p2 p1. h4/path/last p1 = h4/path/first p2 ==> h4/path/first (h4/path/plink p1 p2) = h4/path/first p1
% Assm [h4s_paths_lastu_u_plink]: !p2 p1. h4/path/finite p1 /\ h4/path/finite p2 /\ h4/path/last p1 = h4/path/first p2 ==> h4/path/last (h4/path/plink p1 p2) = h4/path/last p2
% Assm [h4s_paths_okpathu_u_plink]: !p2 p1 R. h4/path/finite p1 /\ h4/path/last p1 = h4/path/first p2 ==> (h4/path/okpath R (h4/path/plink p1 p2) <=> h4/path/okpath R p1 /\ h4/path/okpath R p2)
% Assm [h4s_paths_finiteu_u_plink]: !p2 p1. h4/path/finite (h4/path/plink p1 p2) <=> h4/path/finite p1 /\ h4/path/finite p2
% Assm [h4s_paths_labelsu_u_defu_c0]: !x. h4/path/labels (h4/path/stopped__at x) = h4/llist/LNIL
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_paths_plinku_u_defu_c0]: !x p. h4/path/plink (h4/path/stopped__at x) p = p
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% 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_llists_LCONSu_u_11]: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm [h4s_paths_stoppedu_u_atu_u_notu_u_pconsu_c0]: !y x r p. ~(h4/path/stopped__at x = h4/path/pcons y r p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_paths_tailu_u_def]: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% 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_paths_stoppedu_u_atu_u_11]: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_paths_firstu_u_labelu_u_def]: !x r p. h4/path/first__label (h4/path/pcons x r p) = r
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% 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_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~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_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% 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_paths_lastu_u_thmu_c0]: !x. h4/path/last (h4/path/stopped__at x) = x
% Assm [h4s_paths_lastu_u_thmu_c1]: !x r p. h4/path/last (h4/path/pcons x r p) = h4/path/last p
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_paths_firstu_u_thmu_c1]: !x r p. h4/path/first (h4/path/pcons x r p) = x
% Assm [h4s_paths_firstu_u_thmu_c0]: !x. h4/path/first (h4/path/stopped__at x) = 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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_paths_finiteu_u_pathu_u_ind]: !P. (!x. happ P (h4/path/stopped__at x)) /\ (!x r p. h4/path/finite p /\ happ P p ==> happ P (h4/path/pcons x r p)) ==> (!q. h4/path/finite q ==> happ P q)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% 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_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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_paths_pconcatu_u_def]: !p2 p1 lab. h4/path/pconcat p1 lab p2 = h4/path/toPath (h4/pair/_2C (h4/path/first p1) (h4/llist/LAPPEND (h4/pair/SND (h4/path/fromPath p1)) (h4/llist/LCONS (h4/pair/_2C lab (h4/path/first p2)) (h4/pair/SND (h4/path/fromPath p2)))))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_llists_LAPPENDu_u_EQu_u_LNIL]: !l2 l1. h4/llist/LAPPEND l1 l2 = h4/llist/LNIL <=> l1 = h4/llist/LNIL /\ l2 = h4/llist/LNIL
% Assm [h4s_llists_LCONSu_u_NOTu_u_NILu_c0]: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm [h4s_llists_LFLATTENu_u_APPEND]: !t h. h4/llist/LFLATTEN (h4/llist/LCONS h t) = h4/llist/LAPPEND h (h4/llist/LFLATTEN t)
% Assm [h4s_llists_LTAKEu_u_DROPu_c0]: !n ll. ~h4/llist/LFINITE ll ==> h4/llist/LAPPEND (h4/llist/fromList (h4/option/THE (h4/llist/LTAKE n ll))) (h4/option/THE (h4/llist/LDROP n ll)) = ll
% 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_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% 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_paths_stoppedu_u_atu_u_def]: !x. h4/path/stopped__at x = h4/path/toPath (h4/pair/_2C x h4/llist/LNIL)
% Assm [h4s_llists_LLISTu_u_EQ]: !g f. (!x. happ f x = h4/llist/LNIL /\ happ g x = h4/llist/LNIL \/ (?h y. happ f x = h4/llist/LCONS h (happ f y) /\ happ g x = h4/llist/LCONS h (happ g y))) ==> (!x. happ f x = happ g x)
% Assm [h4s_paths_pathu_u_repu_u_bijectionsu_u_thmu_c1]: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm [h4s_llists_LMAP0u_c1]: !t h f. h4/llist/LMAP f (h4/llist/LCONS h t) = h4/llist/LCONS (happ f h) (h4/llist/LMAP f t)
% Assm [h4s_llists_LMAP0u_c0]: !f. h4/llist/LMAP f h4/llist/LNIL = h4/llist/LNIL
% 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_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_paths_isu_u_stoppedu_u_thmu_c1]: !x r p. h4/path/is__stopped (h4/path/pcons x r p) <=> F
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_paths_isu_u_stoppedu_u_thmu_c0]: !x. h4/path/is__stopped (h4/path/stopped__at x) <=> T
% Assm [h4s_paths_pathu_u_Axiom]: !_4. (!y l g v. happ (happ (happ (happ _4 y) l) g) v = h4/path/pcons y l (happ g v)) ==> (!_3. (!y g l. happ (happ (happ _3 y) g) l = happ (happ (happ _4 y) l) g) ==> (!_2. (!y g v3. happ (happ (happ _2 y) g) v3 = h4/pair/pair__CASE v3 (happ (happ _3 y) g)) ==> (!_1. (!y g v2. happ (happ (happ _1 y) g) v2 = h4/option/option__CASE v2 (h4/path/stopped__at y) (happ (happ _2 y) g)) ==> (!_0. (!g y. happ (happ _0 g) y = happ (happ _1 y) g) ==> (!f. ?g. !x. happ g x = h4/pair/pair__CASE (happ f x) (happ _0 g))))))
% Assm [h4s_paths_pathu_u_bisimulation]: !p2 p1. p1 = p2 <=> (?R. happ (happ R p1) p2 /\ (!q1 q2. happ (happ R q1) q2 ==> (?x. q1 = h4/path/stopped__at x /\ q2 = h4/path/stopped__at x) \/ (?x r q1_27 q2_27. q1 = h4/path/pcons x r q1_27 /\ q2 = h4/path/pcons x r q2_27 /\ happ (happ R q1_27) q2_27)))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_paths_okpathu_u_thmu_c1]: !x r p R. h4/path/okpath R (h4/path/pcons x r p) <=> happ (happ (happ R x) r) (h4/path/first p) /\ h4/path/okpath R p
% Assm [h4s_paths_okpathu_u_thmu_c0]: !x R. h4/path/okpath R (h4/path/stopped__at x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_paths_finiteu_u_thmu_c1]: !x r p. h4/path/finite (h4/path/pcons x r p) <=> h4/path/finite p
% Assm [h4s_paths_finiteu_u_thmu_c0]: !x. h4/path/finite (h4/path/stopped__at x) <=> T
% Assm [h4s_llists_LAPPEND0u_c0]: !x. h4/llist/LAPPEND h4/llist/LNIL x = x
% Assm [h4s_llists_LAPPEND0u_c1]: !x t h. h4/llist/LAPPEND (h4/llist/LCONS h t) x = h4/llist/LCONS h (h4/llist/LAPPEND t x)
% Assm [h4s_llists_llistu_u_CASES]: !l. l = h4/llist/LNIL \/ (?h t. l = h4/llist/LCONS h t)
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_llists_LFLATTENu_u_THMu_c2]: !tl t h. h4/llist/LFLATTEN (h4/llist/LCONS (h4/llist/LCONS h t) tl) = h4/llist/LCONS h (h4/llist/LFLATTEN (h4/llist/LCONS t tl))
% Assm [h4s_llists_LFLATTENu_u_THMu_c1]: !t. h4/llist/LFLATTEN (h4/llist/LCONS h4/llist/LNIL t) = h4/llist/LFLATTEN t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% 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_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_llists_LLISTu_u_STRONGu_u_BISIMULATION]: !ll2 ll1. ll1 = ll2 <=> (?R. happ (happ R ll1) ll2 /\ (!ll3 ll4. happ (happ R ll3) ll4 ==> ll3 = ll4 \/ (?h t1 t2. ll3 = h4/llist/LCONS h t1 /\ ll4 = h4/llist/LCONS h t2 /\ happ (happ R t1) t2)))
% 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_llists_NOTu_u_LFINITEu_u_TAKE]: !ll. ~h4/llist/LFINITE ll ==> (!n. ?y. h4/llist/LTAKE n ll = h4/option/SOME y)
% Assm [h4s_llists_fromList0u_c0]: h4/llist/fromList h4/list/NIL = h4/llist/LNIL
% Assm [h4s_llists_fromList0u_c1]: !t h. h4/llist/fromList (happ (h4/list/CONS h) t) = h4/llist/LCONS h (h4/llist/fromList t)
% Assm [h4s_llists_LFINITEu_u_THMu_c1]: !t h. h4/llist/LFINITE (h4/llist/LCONS h t) <=> h4/llist/LFINITE t
% Assm [h4s_llists_LDROPu_u_THMu_c0]: !ll. h4/llist/LDROP h4/num/0 ll = h4/option/SOME ll
% Assm [h4s_llists_LFINITEu_u_THMu_c0]: h4/llist/LFINITE h4/llist/LNIL <=> T
% Assm [h4s_llists_LDROPu_u_THMu_c2]: !t n h. h4/llist/LDROP (h4/num/SUC n) (h4/llist/LCONS h t) = h4/llist/LDROP n t
% Assm [h4s_llists_LTAKEu_u_THMu_c2]: !t n h. h4/llist/LTAKE (h4/num/SUC n) (h4/llist/LCONS h t) = h4/option/OPTION__MAP (h4/list/CONS h) (h4/llist/LTAKE n t)
% Assm [h4s_llists_LTAKEu_u_THMu_c0]: !l. h4/llist/LTAKE h4/num/0 l = h4/option/SOME h4/list/NIL
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (h4/option/SOME x) = x
% 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_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Goal: !p2 p1. h4/path/labels (h4/path/plink p1 p2) = h4/llist/LAPPEND (h4/path/labels p1) (h4/path/labels p2)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1230491,TV_Q1230487]: ![V_f, V_g]: (![V_x]: s(TV_Q1230487,happ(s(t_fun(TV_Q1230491,TV_Q1230487),V_f),s(TV_Q1230491,V_x))) = s(TV_Q1230487,happ(s(t_fun(TV_Q1230491,TV_Q1230487),V_g),s(TV_Q1230491,V_x))) => s(t_fun(TV_Q1230491,TV_Q1230487),V_f) = s(t_fun(TV_Q1230491,TV_Q1230487),V_g))).
fof(ah4s_paths_labelsu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(t_h4s_llists_llist(TV_u_27b),h4s_paths_labels(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_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_r),s(t_h4s_llists_llist(TV_u_27b),h4s_paths_labels(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
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),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_paths_plinku_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p2, V_p1]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(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_p1))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_paths_firstu_u_plink, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1]: (s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))) => s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))))).
fof(ah4s_paths_lastu_u_plink, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1]: ((p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1)))) & (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))) & s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))))) => s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))))) = s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))))).
fof(ah4s_paths_okpathu_u_plink, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1, V_R]: ((p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1)))) & s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))) => (p(s(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_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))))) <=> (p(s(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_p1)))) & p(s(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_p2)))))))).
fof(ah4s_paths_finiteu_u_plink, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1]: (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))))) <=> (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1)))) & p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))))))).
fof(ah4s_paths_labelsu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_llists_llist(TV_u_27b),h4s_paths_labels(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_llists_llist(TV_u_27b),h4s_llists_lnil)).
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_plinku_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(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),V_p))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_paths_stoppedu_u_atu_u_notu_u_pconsu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_r, V_p]: ~ (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_y),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_paths_tailu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)).
fof(ah4s_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_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_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_paths_firstu_u_labelu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(TV_u_27b,h4s_paths_firstu_u_label(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27b,V_r)).
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_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_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_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_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_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_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_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_paths_lastu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_paths_lastu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_paths_firstu_u_thmu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_r, V_p]: s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27a,V_x)).
fof(ah4s_paths_firstu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_paths_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_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_paths_finiteu_u_pathu_u_ind, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x)))))) & ![V_x, V_r, V_p]: ((p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) & p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) => p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))) => ![V_q]: (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q)))) => p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))))).
fof(ah4s_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_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_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_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_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_paths_pconcatu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1, V_lab]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pconcat(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(TV_u_27b,V_lab),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_pairs_u_2c(s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_llists_lappend(s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_paths_frompath(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))))),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_llists_lcons(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_lab),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))))),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_paths_frompath(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))))))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_llists_LAPPENDu_u_EQu_u_LNIL, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),V_l1),s(t_h4s_llists_llist(TV_u_27a),V_l2))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil) <=> (s(t_h4s_llists_llist(TV_u_27a),V_l1) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil) & s(t_h4s_llists_llist(TV_u_27a),V_l2) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)))).
fof(ah4s_llists_LCONSu_u_NOTu_u_NILu_c0, axiom, ![TV_u_27a]: ![V_t, V_h]: ~ (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))).
fof(ah4s_llists_LFLATTENu_u_APPEND, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lflatten(s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27a)),h4s_llists_lcons(s(t_h4s_llists_llist(TV_u_27a),V_h),s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27a)),V_t))))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),V_h),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lflatten(s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27a)),V_t)))))).
fof(ah4s_llists_LTAKEu_u_DROPu_c0, axiom, ![TV_u_27a]: ![V_n, V_ll]: (~ (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll))))) => s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_fromlist(s(t_h4s_lists_list(TV_u_27a),h4s_options_the(s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27a),V_ll))))))),s(t_h4s_llists_llist(TV_u_27a),h4s_options_the(s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ldrop(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27a),V_ll))))))) = s(t_h4s_llists_llist(TV_u_27a),V_ll))).
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_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_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_llists_LLISTu_u_EQ, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (![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_f),s(TV_u_27a,V_x))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil) & 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_llists_lnil)) | ?[V_h, V_y]: (s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(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_h),s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_f),s(TV_u_27a,V_y))))) & 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_llists_lcons(s(TV_u_27b,V_h),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_y))))))) => ![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_f),s(TV_u_27a,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))))).
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_llists_LMAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(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),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_llists_LMAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(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_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_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_paths_isu_u_stoppedu_u_thmu_c1, axiom, ![TV_u_27c,TV_u_27d]: ![V_x, V_r, V_p]: s(t_bool,h4s_paths_isu_u_stopped(s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_pcons(s(TV_u_27c,V_x),s(TV_u_27d,V_r),s(t_h4s_paths_path(TV_u_27c,TV_u_27d),V_p))))) = s(t_bool,f)).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_paths_isu_u_stoppedu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_bool,h4s_paths_isu_u_stopped(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_paths_pathu_u_Axiom, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_4]: (![V_y, V_l, V_g, V_v]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c))))),V_uu_4),s(TV_u_27b,V_y))),s(TV_u_27c,V_l))),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))),s(TV_u_27a,V_v))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pcons(s(TV_u_27b,V_y),s(TV_u_27c,V_l),s(t_h4s_paths_path(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g),s(TV_u_27a,V_v))))) => ![V_uu_3]: (![V_y, V_g, V_l]: s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c))))),V_uu_3),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))),s(TV_u_27c,V_l))) = s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c))))),V_uu_4),s(TV_u_27b,V_y))),s(TV_u_27c,V_l))),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))) => ![V_uu_2]: (![V_y, V_g, V_v3]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_paths_path(TV_u_27b,TV_u_27c)))),V_uu_2),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))),s(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),V_v3))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),V_v3),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c))))),V_uu_3),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))))) => ![V_uu_1]: (![V_y, V_g, V_v2]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),happ(s(t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c)))),V_uu_1),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))),s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),V_v2))) = 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_27c,TV_u_27a)),V_v2),s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_stoppedu_u_at(s(TV_u_27b,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_paths_path(TV_u_27b,TV_u_27c)))),V_uu_2),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))))) => ![V_uu_0]: (![V_g, V_y]: s(t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27b,t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))),s(TV_u_27b,V_y))) = s(t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c)))),V_uu_1),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))) => ![V_f]: ?[V_g]: ![V_x]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g),s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)))),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),t_fun(TV_u_27b,t_fun(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_h4s_paths_path(TV_u_27b,TV_u_27c)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_paths_path(TV_u_27b,TV_u_27c)),V_g))))))))))).
fof(ah4s_paths_pathu_u_bisimulation, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2) <=> ?[V_R]: (p(s(t_bool,happ(s(t_fun(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_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),V_R),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))) & ![V_q1, V_q2]: (p(s(t_bool,happ(s(t_fun(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_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),V_R),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q1))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q2)))) => (?[V_x]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q1) = 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),V_q2) = 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_q1u_27, V_q2u_27]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q1) = 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_q1u_27))) & (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q2) = 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_q2u_27))) & p(s(t_bool,happ(s(t_fun(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_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool)),V_R),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q1u_27))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q2u_27))))))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_paths_okpathu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p, V_R]: (p(s(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,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_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_R]: p(s(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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_paths_finiteu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(t_bool,h4s_paths_finite(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,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))).
fof(ah4s_paths_finiteu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_llists_LAPPEND0u_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil),s(t_h4s_llists_llist(TV_u_27a),V_x))) = s(t_h4s_llists_llist(TV_u_27a),V_x)).
fof(ah4s_llists_LAPPEND0u_c1, axiom, ![TV_u_27a]: ![V_x, V_t, V_h]: s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))),s(t_h4s_llists_llist(TV_u_27a),V_x))) = 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),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),V_t),s(t_h4s_llists_llist(TV_u_27a),V_x)))))).
fof(ah4s_llists_llistu_u_CASES, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_llists_llist(TV_u_27a),V_l) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil) | ?[V_h, V_t]: s(t_h4s_llists_llist(TV_u_27a),V_l) = 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))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_llists_LFLATTENu_u_THMu_c2, axiom, ![TV_u_27d]: ![V_tl, V_t, V_h]: s(t_h4s_llists_llist(TV_u_27d),h4s_llists_lflatten(s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27d)),h4s_llists_lcons(s(t_h4s_llists_llist(TV_u_27d),h4s_llists_lcons(s(TV_u_27d,V_h),s(t_h4s_llists_llist(TV_u_27d),V_t))),s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27d)),V_tl))))) = s(t_h4s_llists_llist(TV_u_27d),h4s_llists_lcons(s(TV_u_27d,V_h),s(t_h4s_llists_llist(TV_u_27d),h4s_llists_lflatten(s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27d)),h4s_llists_lcons(s(t_h4s_llists_llist(TV_u_27d),V_t),s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27d)),V_tl)))))))).
fof(ah4s_llists_LFLATTENu_u_THMu_c1, axiom, ![TV_u_27c]: ![V_t]: s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lflatten(s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27c)),h4s_llists_lcons(s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lnil),s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27c)),V_t))))) = s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lflatten(s(t_h4s_llists_llist(t_h4s_llists_llist(TV_u_27c)),V_t)))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_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_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_llists_LLISTu_u_STRONGu_u_BISIMULATION, axiom, ![TV_u_27a]: ![V_ll2, V_ll1]: (s(t_h4s_llists_llist(TV_u_27a),V_ll1) = s(t_h4s_llists_llist(TV_u_27a),V_ll2) <=> ?[V_R]: (p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_llists_llist(TV_u_27a),t_bool)),V_R),s(t_h4s_llists_llist(TV_u_27a),V_ll1))),s(t_h4s_llists_llist(TV_u_27a),V_ll2)))) & ![V_ll3, V_ll4]: (p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_llists_llist(TV_u_27a),t_bool)),V_R),s(t_h4s_llists_llist(TV_u_27a),V_ll3))),s(t_h4s_llists_llist(TV_u_27a),V_ll4)))) => (s(t_h4s_llists_llist(TV_u_27a),V_ll3) = s(t_h4s_llists_llist(TV_u_27a),V_ll4) | ?[V_h, V_t1, V_t2]: (s(t_h4s_llists_llist(TV_u_27a),V_ll3) = 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_t1))) & (s(t_h4s_llists_llist(TV_u_27a),V_ll4) = 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_t2))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_llists_llist(TV_u_27a),t_bool)),V_R),s(t_h4s_llists_llist(TV_u_27a),V_t1))),s(t_h4s_llists_llist(TV_u_27a),V_t2))))))))))).
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_llists_NOTu_u_LFINITEu_u_TAKE, axiom, ![TV_u_27a]: ![V_ll]: (~ (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll))))) => ![V_n]: ?[V_y]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_some(s(t_h4s_lists_list(TV_u_27a),V_y))))).
fof(ah4s_llists_fromList0u_c0, axiom, ![TV_u_27a]: s(t_h4s_llists_llist(TV_u_27a),h4s_llists_fromlist(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)).
fof(ah4s_llists_fromList0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_llists_llist(TV_u_27a),h4s_llists_fromlist(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_cons(s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = 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),h4s_llists_fromlist(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_llists_LFINITEu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_bool,h4s_llists_lfinite(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_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27b),V_t)))).
fof(ah4s_llists_LDROPu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_ll]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ldrop(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_options_some(s(t_h4s_llists_llist(TV_u_27a),V_ll)))).
fof(ah4s_llists_LFINITEu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_bool,t)).
fof(ah4s_llists_LDROPu_u_THMu_c2, axiom, ![TV_u_27c]: ![V_t, V_n, V_h]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27c)),h4s_llists_ldrop(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lcons(s(TV_u_27c,V_h),s(t_h4s_llists_llist(TV_u_27c),V_t))))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27c)),h4s_llists_ldrop(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27c),V_t)))).
fof(ah4s_llists_LTAKEu_u_THMu_c2, axiom, ![TV_u_27c]: ![V_t, V_n, V_h]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27c)),h4s_llists_ltake(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lcons(s(TV_u_27c,V_h),s(t_h4s_llists_llist(TV_u_27c),V_t))))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27c)),h4s_options_optionu_u_map(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c)),h4s_lists_cons(s(TV_u_27c,V_h))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27c)),h4s_llists_ltake(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27c),V_t)))))).
fof(ah4s_llists_LTAKEu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_llists_llist(TV_u_27a),V_l))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_some(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_options_THEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_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_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ch4s_paths_labelsu_u_plink, conjecture, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1]: s(t_h4s_llists_llist(TV_u_27b),h4s_paths_labels(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27b),h4s_paths_labels(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))),s(t_h4s_llists_llist(TV_u_27b),h4s_paths_labels(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))))).
