%   ORIGINAL: h4/llist/LFINITE__fromList
% 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/list/list__Axiom: !f1 f0. ?fn. fn h4/list/NIL = f0 /\ (!a0 a1. fn (h4/list/CONS a0 a1) = f1 a0 a1 (fn a1))
% 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/fromList0_c0: h4/llist/fromList h4/list/NIL = h4/llist/LNIL
% Assm: h4/llist/LFINITE__def: h4/llist/LFINITE = (\a0. !LFINITE_27. (!a00. a00 = h4/llist/LNIL \/ (?h t. a00 = h4/llist/LCONS h t /\ LFINITE_27 t) ==> LFINITE_27 a00) ==> LFINITE_27 a0)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/llist/LFINITE__THM_c0: h4/llist/LFINITE h4/llist/LNIL <=> T
% Assm: h4/llist/LFINITE__THM_c1: !t h. h4/llist/LFINITE (h4/llist/LCONS h t) <=> h4/llist/LFINITE t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/llist/LFINITE__INDUCTION: !P. P h4/llist/LNIL /\ (!h t. P t ==> P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> P a0)
% Assm: h4/llist/LLENGTH0: !ll. h4/llist/LLENGTH ll = h4/bool/COND (h4/llist/LFINITE ll) (h4/option/SOME (h4/min/_40 (\n. h4/llist/llength__rel ll n))) h4/option/NONE
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/llist/LFINITE__STRONG__INDUCTION: !P. P h4/llist/LNIL /\ (!h t. h4/llist/LFINITE t /\ P t ==> P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> P a0)
% Assm: h4/llist/NOT__LFINITE__NO__LENGTH: !ll. ~h4/llist/LFINITE ll ==> h4/llist/LLENGTH ll = h4/option/NONE
% Assm: h4/llist/LFINITE__HAS__LENGTH: !ll. h4/llist/LFINITE ll ==> (?n. h4/llist/LLENGTH ll = h4/option/SOME n)
% Assm: h4/llist/LFINITE__APPEND: !ll2 ll1. h4/llist/LFINITE (h4/llist/LAPPEND ll1 ll2) <=> h4/llist/LFINITE ll1 /\ h4/llist/LFINITE ll2
% Assm: h4/llist/LFINITE0: !ll. h4/llist/LFINITE ll <=> (?n. h4/llist/LTAKE n ll = h4/option/NONE)
% Assm: h4/llist/toList0: !ll. h4/llist/toList ll = h4/bool/COND (h4/llist/LFINITE ll) (h4/llist/LTAKE (h4/option/THE (h4/llist/LLENGTH ll)) ll) h4/option/NONE
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/llist/LFINITE__MAP: !ll f. h4/llist/LFINITE (h4/llist/LMAP f ll) <=> h4/llist/LFINITE ll
% Assm: h4/llist/LFINITE__ind: !LFINITE_27. LFINITE_27 h4/llist/LNIL /\ (!h t. LFINITE_27 t ==> LFINITE_27 (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> LFINITE_27 a0)
% Assm: h4/llist/NOT__LFINITE__APPEND: !ll2 ll1. ~h4/llist/LFINITE ll1 ==> h4/llist/LAPPEND ll1 ll2 = ll1
% Assm: h4/llist/LFINITE__rules_c1: !t h. h4/llist/LFINITE t ==> h4/llist/LFINITE (h4/llist/LCONS h t)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/llist/LLENGTH__APPEND: !ll2 ll1. h4/llist/LLENGTH (h4/llist/LAPPEND ll1 ll2) = h4/bool/COND (h4/llist/LFINITE ll1 /\ h4/llist/LFINITE ll2) (h4/option/SOME (h4/arithmetic/_2B (h4/option/THE (h4/llist/LLENGTH ll1)) (h4/option/THE (h4/llist/LLENGTH ll2)))) h4/option/NONE
% Assm: h4/llist/LFINITE__rules_c0: h4/llist/LFINITE h4/llist/LNIL
% Assm: h4/llist/LFINITE__cases: !a0. h4/llist/LFINITE a0 <=> a0 = h4/llist/LNIL \/ (?h t. a0 = h4/llist/LCONS h t /\ h4/llist/LFINITE t)
% Assm: h4/llist/LFINITE__strongind: !LFINITE_27. LFINITE_27 h4/llist/LNIL /\ (!h t. h4/llist/LFINITE t /\ LFINITE_27 t ==> LFINITE_27 (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> LFINITE_27 a0)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/llist/LCONS__NOT__NIL_c0: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% 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/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% 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/llist/llist__CASES: !l. l = h4/llist/LNIL \/ (?h t. l = h4/llist/LCONS h t)
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% 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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/bool/IMP__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/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% 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/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/llist/LMAP0_c0: !f. h4/llist/LMAP f h4/llist/LNIL = h4/llist/LNIL
% 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/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> 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/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/llist/LAPPEND0_c0: !x. h4/llist/LAPPEND h4/llist/LNIL x = x
% Assm: h4/llist/LTAKE__NIL__EQ__NONE: !m. h4/llist/LTAKE m h4/llist/LNIL = h4/option/NONE <=> h4/prim__rec/_3C h4/num/0 m
% Assm: h4/llist/LTAKE__THM_c0: !l. h4/llist/LTAKE h4/num/0 l = h4/option/SOME h4/list/NIL
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% 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/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% 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/LLIST__BISIMULATION: !ll2 ll1. ll1 = ll2 <=> (?R. R ll1 ll2 /\ (!ll3 ll4. R ll3 ll4 ==> ll3 = h4/llist/LNIL /\ ll4 = h4/llist/LNIL \/ h4/llist/LHD ll3 = h4/llist/LHD ll4 /\ R (h4/option/THE (h4/llist/LTL ll3)) (h4/option/THE (h4/llist/LTL ll4))))
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P 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/llist/LTAKE0_c1: !n ll. h4/llist/LTAKE (h4/num/SUC n) ll = h4/option/option__CASE (h4/llist/LHD ll) h4/option/NONE (\hd. h4/option/option__CASE (h4/llist/LTAKE n (h4/option/THE (h4/llist/LTL ll))) h4/option/NONE (\tl. h4/option/SOME (h4/list/CONS hd tl)))
% Assm: h4/llist/LLENGTH__THM_c1: !t h. h4/llist/LLENGTH (h4/llist/LCONS h t) = h4/option/OPTION__MAP h4/num/SUC (h4/llist/LLENGTH t)
% Assm: h4/llist/LLENGTH__THM_c0: h4/llist/LLENGTH h4/llist/LNIL = h4/option/SOME h4/num/0
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__CLAUSES_c2: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/llist/LTAKE__EQ__SOME__CONS: !x n l. h4/llist/LTAKE n l = h4/option/SOME x ==> (!h. ?y. h4/llist/LTAKE n (h4/llist/LCONS h l) = h4/option/SOME y)
% Assm: h4/llist/LTAKE__NIL__EQ__SOME: !m l. h4/llist/LTAKE m h4/llist/LNIL = h4/option/SOME l <=> m = h4/num/0 /\ l = h4/list/NIL
% Assm: h4/llist/LTAKE__SNOC__LNTH: !n ll. h4/llist/LTAKE (h4/num/SUC n) ll = h4/option/option__CASE (h4/llist/LTAKE n ll) h4/option/NONE (\l. h4/option/option__CASE (h4/llist/LNTH n ll) h4/option/NONE (\e. h4/option/SOME (h4/list/APPEND l (h4/list/CONS e h4/list/NIL))))
% 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/llist/LAPPEND__EQ__LNIL: !l2 l1. h4/llist/LAPPEND l1 l2 = h4/llist/LNIL <=> l1 = h4/llist/LNIL /\ l2 = h4/llist/LNIL
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/llist/LTAKE0_c0: !ll. h4/llist/LTAKE h4/num/0 ll = h4/option/SOME h4/list/NIL
% Assm: h4/llist/LTL__THM_c0: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/LHD__THM_c0: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% 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/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% 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/llist/LTAKE__THM_c1: !n. h4/llist/LTAKE (h4/num/SUC n) h4/llist/LNIL = h4/option/NONE
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/llist/LNTH__THM_c0: !n. h4/llist/LNTH n h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/LNTH0_c0: !ll. h4/llist/LNTH h4/num/0 ll = h4/llist/LHD ll
% Assm: h4/list/APPEND__11_c0: !l3 l2 l1. h4/list/APPEND l1 l2 = h4/list/APPEND l1 l3 <=> l2 = l3
% Assm: h4/llist/LNTH__THM_c2: !t n h. h4/llist/LNTH (h4/num/SUC n) (h4/llist/LCONS h t) = h4/llist/LNTH n t
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Goal: !l. h4/llist/LFINITE (h4/llist/fromList l)
%   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_lists_listu_u_Axiom]: !f1 f0. ?fn. happ fn h4/list/NIL = f0 /\ (!a0 a1. happ fn (happ (h4/list/CONS a0) a1) = happ (happ (happ f1 a0) a1) (happ fn a1))
% 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_fromList0u_c0]: h4/llist/fromList h4/list/NIL = h4/llist/LNIL
% Assm [h4s_llists_LFINITEu_u_def]: !x. h4/llist/LFINITE x <=> (!LFINITE_27. (!a00. a00 = h4/llist/LNIL \/ (?h t. a00 = h4/llist/LCONS h t /\ happ LFINITE_27 t) ==> happ LFINITE_27 a00) ==> happ LFINITE_27 x)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_llists_LFINITEu_u_THMu_c0]: h4/llist/LFINITE h4/llist/LNIL <=> T
% Assm [h4s_llists_LFINITEu_u_THMu_c1]: !t h. h4/llist/LFINITE (h4/llist/LCONS h t) <=> h4/llist/LFINITE t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_llists_LFINITEu_u_INDUCTION]: !P. happ P h4/llist/LNIL /\ (!h t. happ P t ==> happ P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> happ P a0)
% Assm [h4s_llists_LLENGTH0]: !_0. (!ll n. happ (happ _0 ll) n <=> h4/llist/llength__rel ll n) ==> (!ll. h4/llist/LLENGTH ll = h4/bool/COND (h4/llist/LFINITE ll) (h4/option/SOME (h4/min/_40 (happ _0 ll))) h4/option/NONE)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_llists_LFINITEu_u_STRONGu_u_INDUCTION]: !P. happ P h4/llist/LNIL /\ (!h t. h4/llist/LFINITE t /\ happ P t ==> happ P (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> happ P a0)
% Assm [h4s_llists_NOTu_u_LFINITEu_u_NOu_u_LENGTH]: !ll. ~h4/llist/LFINITE ll ==> h4/llist/LLENGTH ll = h4/option/NONE
% Assm [h4s_llists_LFINITEu_u_HASu_u_LENGTH]: !ll. h4/llist/LFINITE ll ==> (?n. h4/llist/LLENGTH ll = h4/option/SOME n)
% Assm [h4s_llists_LFINITEu_u_APPEND]: !ll2 ll1. h4/llist/LFINITE (h4/llist/LAPPEND ll1 ll2) <=> h4/llist/LFINITE ll1 /\ h4/llist/LFINITE ll2
% Assm [h4s_llists_LFINITE0]: !ll. h4/llist/LFINITE ll <=> (?n. h4/llist/LTAKE n ll = h4/option/NONE)
% Assm [h4s_llists_toList0]: !ll. h4/llist/toList ll = h4/bool/COND (h4/llist/LFINITE ll) (h4/llist/LTAKE (h4/option/THE (h4/llist/LLENGTH ll)) ll) h4/option/NONE
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_llists_LFINITEu_u_MAP]: !ll f. h4/llist/LFINITE (h4/llist/LMAP f ll) <=> h4/llist/LFINITE ll
% Assm [h4s_llists_LFINITEu_u_ind]: !LFINITE_27. happ LFINITE_27 h4/llist/LNIL /\ (!h t. happ LFINITE_27 t ==> happ LFINITE_27 (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> happ LFINITE_27 a0)
% Assm [h4s_llists_NOTu_u_LFINITEu_u_APPEND]: !ll2 ll1. ~h4/llist/LFINITE ll1 ==> h4/llist/LAPPEND ll1 ll2 = ll1
% Assm [h4s_llists_LFINITEu_u_rulesu_c1]: !t h. h4/llist/LFINITE t ==> h4/llist/LFINITE (h4/llist/LCONS h t)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_llists_LLENGTHu_u_APPEND]: !ll2 ll1. ?v. (v <=> h4/llist/LFINITE ll1 /\ h4/llist/LFINITE ll2) /\ h4/llist/LLENGTH (h4/llist/LAPPEND ll1 ll2) = h4/bool/COND v (h4/option/SOME (h4/arithmetic/_2B (h4/option/THE (h4/llist/LLENGTH ll1)) (h4/option/THE (h4/llist/LLENGTH ll2)))) h4/option/NONE
% Assm [h4s_llists_LFINITEu_u_rulesu_c0]: h4/llist/LFINITE h4/llist/LNIL
% Assm [h4s_llists_LFINITEu_u_cases]: !a0. h4/llist/LFINITE a0 <=> a0 = h4/llist/LNIL \/ (?h t. a0 = h4/llist/LCONS h t /\ h4/llist/LFINITE t)
% Assm [h4s_llists_LFINITEu_u_strongind]: !LFINITE_27. happ LFINITE_27 h4/llist/LNIL /\ (!h t. h4/llist/LFINITE t /\ happ LFINITE_27 t ==> happ LFINITE_27 (h4/llist/LCONS h t)) ==> (!a0. h4/llist/LFINITE a0 ==> happ LFINITE_27 a0)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_llists_LCONSu_u_NOTu_u_NILu_c0]: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% 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_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% 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_llists_llistu_u_CASES]: !l. l = h4/llist/LNIL \/ (?h t. l = h4/llist/LCONS h t)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !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_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% 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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (happ h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_bools_IMPu_u_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_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% 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_options_THEu_u_DEF]: !x. h4/option/THE (h4/option/SOME x) = x
% Assm [h4s_llists_LMAP0u_c0]: !f. h4/llist/LMAP f h4/llist/LNIL = h4/llist/LNIL
% 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_bools_LEFTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. happ P x ==> Q) <=> (?x. happ P x) ==> Q
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_llists_LTAKEu_u_THMu_c2]: !t n h. h4/llist/LTAKE (happ h4/num/SUC n) (h4/llist/LCONS h t) = h4/option/OPTION__MAP (h4/list/CONS h) (h4/llist/LTAKE n t)
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (happ h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_llists_LAPPEND0u_c0]: !x. h4/llist/LAPPEND h4/llist/LNIL x = x
% Assm [h4s_llists_LTAKEu_u_NILu_u_EQu_u_NONE]: !m. h4/llist/LTAKE m h4/llist/LNIL = h4/option/NONE <=> h4/prim__rec/_3C h4/num/0 m
% Assm [h4s_llists_LTAKEu_u_THMu_c0]: !l. h4/llist/LTAKE h4/num/0 l = h4/option/SOME h4/list/NIL
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (happ h4/num/SUC n)
% 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_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% 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_LLISTu_u_BISIMULATION]: !ll2 ll1. ll1 = ll2 <=> (?R. happ (happ R ll1) ll2 /\ (!ll3 ll4. happ (happ R ll3) ll4 ==> ll3 = h4/llist/LNIL /\ ll4 = h4/llist/LNIL \/ h4/llist/LHD ll3 = h4/llist/LHD ll4 /\ happ (happ R (h4/option/THE (h4/llist/LTL ll3))) (h4/option/THE (h4/llist/LTL ll4))))
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P 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_llists_LTAKE0u_c1]: !_1. (!hd tl. happ (happ _1 hd) tl = h4/option/SOME (happ (h4/list/CONS hd) tl)) ==> (!_0. (!n ll hd. happ (happ (happ _0 n) ll) hd = h4/option/option__CASE (h4/llist/LTAKE n (h4/option/THE (h4/llist/LTL ll))) h4/option/NONE (happ _1 hd)) ==> (!n ll. h4/llist/LTAKE (happ h4/num/SUC n) ll = h4/option/option__CASE (h4/llist/LHD ll) h4/option/NONE (happ (happ _0 n) ll)))
% Assm [h4s_llists_LLENGTHu_u_THMu_c1]: !t h. h4/llist/LLENGTH (h4/llist/LCONS h t) = h4/option/OPTION__MAP h4/num/SUC (h4/llist/LLENGTH t)
% Assm [h4s_llists_LLENGTHu_u_THMu_c0]: h4/llist/LLENGTH h4/llist/LNIL = h4/option/SOME h4/num/0
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c2]: !n m. h4/arithmetic/_2B (happ h4/num/SUC m) n = happ h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. happ h4/num/SUC m = happ h4/num/SUC n <=> m = n
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_llists_LTAKEu_u_EQu_u_SOMEu_u_CONS]: !x n l. h4/llist/LTAKE n l = h4/option/SOME x ==> (!h. ?y. h4/llist/LTAKE n (h4/llist/LCONS h l) = h4/option/SOME y)
% Assm [h4s_llists_LTAKEu_u_NILu_u_EQu_u_SOME]: !m l. h4/llist/LTAKE m h4/llist/LNIL = h4/option/SOME l <=> m = h4/num/0 /\ l = h4/list/NIL
% Assm [h4s_llists_LTAKEu_u_SNOCu_u_LNTH]: !_1. (!l e. happ (happ _1 l) e = h4/option/SOME (h4/list/APPEND l (happ (h4/list/CONS e) h4/list/NIL))) ==> (!_0. (!n ll l. happ (happ (happ _0 n) ll) l = h4/option/option__CASE (h4/llist/LNTH n ll) h4/option/NONE (happ _1 l)) ==> (!n ll. h4/llist/LTAKE (happ h4/num/SUC n) ll = h4/option/option__CASE (h4/llist/LTAKE n ll) h4/option/NONE (happ (happ _0 n) ll)))
% 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_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_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_llists_LTAKE0u_c0]: !ll. h4/llist/LTAKE h4/num/0 ll = h4/option/SOME h4/list/NIL
% Assm [h4s_llists_LTLu_u_THMu_c0]: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_LHDu_u_THMu_c0]: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% 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_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% 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_llists_LTAKEu_u_THMu_c1]: !n. h4/llist/LTAKE (happ h4/num/SUC n) h4/llist/LNIL = h4/option/NONE
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = happ h4/num/SUC n)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(happ h4/num/SUC n = h4/num/0)
% Assm [h4s_llists_LNTHu_u_THMu_c0]: !n. h4/llist/LNTH n h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_LNTH0u_c0]: !ll. h4/llist/LNTH h4/num/0 ll = h4/llist/LHD ll
% Assm [h4s_lists_APPENDu_u_11u_c0]: !l3 l2 l1. h4/list/APPEND l1 l2 = h4/list/APPEND l1 l3 <=> l2 = l3
% Assm [h4s_llists_LNTHu_u_THMu_c2]: !t n h. h4/llist/LNTH (happ h4/num/SUC n) (h4/llist/LCONS h t) = h4/llist/LNTH n t
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (happ (h4/list/CONS h) l1) l2 = happ (h4/list/CONS h) (h4/list/APPEND l1 l2)
% Goal: !l. h4/llist/LFINITE (h4/llist/fromList l)
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_Q1259007,TV_Q1259003]: ![V_f, V_g]: (![V_x]: s(TV_Q1259003,happ(s(t_fun(TV_Q1259007,TV_Q1259003),V_f),s(TV_Q1259007,V_x))) = s(TV_Q1259003,happ(s(t_fun(TV_Q1259007,TV_Q1259003),V_g),s(TV_Q1259007,V_x))) => s(t_fun(TV_Q1259007,TV_Q1259003),V_f) = s(t_fun(TV_Q1259007,TV_Q1259003),V_g))).
fof(ah4s_lists_listu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f1, V_f0]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_f0) & ![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),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_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b))),V_f1),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),V_a1))))))).
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_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_LFINITEu_u_def, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_x)))) <=> ![V_LFINITEu_27]: (![V_a00]: ((s(t_h4s_llists_llist(TV_u_27a),V_a00) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil) | ?[V_h, V_t]: (s(t_h4s_llists_llist(TV_u_27a),V_a00) = 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))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),V_t)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),V_x))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
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_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_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_llists_LFINITEu_u_INDUCTION, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)))) & ![V_h, V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_t)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t)))))))) => ![V_a0]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_a0))))))).
fof(ah4s_llists_LLENGTH0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_ll, V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_llists_llengthu_u_rel(s(t_h4s_llists_llist(TV_u_27a),V_ll),s(t_h4s_nums_num,V_n))) => ![V_ll]: s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_mins_u_40(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_llists_LFINITEu_u_STRONGu_u_INDUCTION, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)))) & ![V_h, V_t]: ((p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_t)))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_t))))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t)))))))) => ![V_a0]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_P),s(t_h4s_llists_llist(TV_u_27a),V_a0))))))).
fof(ah4s_llists_NOTu_u_LFINITEu_u_NOu_u_LENGTH, axiom, ![TV_u_27a]: ![V_ll]: (~ (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll))))) => s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))).
fof(ah4s_llists_LFINITEu_u_HASu_u_LENGTH, 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]: s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_n))))).
fof(ah4s_llists_LFINITEu_u_APPEND, axiom, ![TV_u_27a]: ![V_ll2, V_ll1]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),V_ll1),s(t_h4s_llists_llist(TV_u_27a),V_ll2)))))) <=> (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll1)))) & p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll2))))))).
fof(ah4s_llists_LFINITE0, 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]: 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_none))).
fof(ah4s_llists_toList0, axiom, ![TV_u_27a]: ![V_ll]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_tolist(s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_bools_cond(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,h4s_options_the(s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll))))),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none)))).
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_llists_LFINITEu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_ll, V_f]: s(t_bool,h4s_llists_lfinite(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_ll))))) = s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll)))).
fof(ah4s_llists_LFINITEu_u_ind, axiom, ![TV_u_27a]: ![V_LFINITEu_27]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)))) & ![V_h, V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),V_t)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t)))))))) => ![V_a0]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),V_a0))))))).
fof(ah4s_llists_NOTu_u_LFINITEu_u_APPEND, axiom, ![TV_u_27a]: ![V_ll2, V_ll1]: (~ (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll1))))) => s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),V_ll1),s(t_h4s_llists_llist(TV_u_27a),V_ll2))) = s(t_h4s_llists_llist(TV_u_27a),V_ll1))).
fof(ah4s_llists_LFINITEu_u_rulesu_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_t)))) => p(s(t_bool,h4s_llists_lfinite(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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_llists_LLENGTHu_u_APPEND, axiom, ![TV_u_27a]: ![V_ll2, V_ll1]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll1)))) & p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll2)))))) & s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lappend(s(t_h4s_llists_llist(TV_u_27a),V_ll1),s(t_h4s_llists_llist(TV_u_27a),V_ll2))))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_options_the(s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll1))))),s(t_h4s_nums_num,h4s_options_the(s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll2))))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))).
fof(ah4s_llists_LFINITEu_u_rulesu_c0, axiom, ![TV_u_27a]: p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))))).
fof(ah4s_llists_LFINITEu_u_cases, axiom, ![TV_u_27a]: ![V_a0]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_a0)))) <=> (s(t_h4s_llists_llist(TV_u_27a),V_a0) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil) | ?[V_h, V_t]: (s(t_h4s_llists_llist(TV_u_27a),V_a0) = 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))) & p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_t)))))))).
fof(ah4s_llists_LFINITEu_u_strongind, axiom, ![TV_u_27a]: ![V_LFINITEu_27]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)))) & ![V_h, V_t]: ((p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_t)))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),V_t))))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t)))))))) => ![V_a0]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_a0)))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_LFINITEu_27),s(t_h4s_llists_llist(TV_u_27a),V_a0))))))).
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_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_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_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_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_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_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_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_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_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_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_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_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_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_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_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_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_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bools_IMPu_u_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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_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_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_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_bools_LEFTu_u_FORALLu_u_IMPu_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_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_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_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_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
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_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_LTAKEu_u_NILu_u_EQu_u_NONE, axiom, ![TV_u_27a]: ![V_m]: (s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,V_m),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m)))))).
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_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_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_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_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_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_LLISTu_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),h4s_llists_lnil) & s(t_h4s_llists_llist(TV_u_27a),V_ll4) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)) | (s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),V_ll3))) = s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),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),h4s_options_the(s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27a),V_ll3))))))),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_ltl(s(t_h4s_llists_llist(TV_u_27a),V_ll4)))))))))))))).
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_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_llists_LTAKE0u_c1, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_hd, V_tl]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(TV_u_27a,V_hd))),s(t_h4s_lists_list(TV_u_27a),V_tl))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_some(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_hd))),s(t_h4s_lists_list(TV_u_27a),V_tl))))) => ![V_uu_0]: (![V_n, V_ll, V_hd]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(TV_u_27a,V_hd))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_optionu_u_case(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),h4s_options_the(s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27a),V_ll))))))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(TV_u_27a,V_hd))))) => ![V_n, V_ll]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll)))))))).
fof(ah4s_llists_LLENGTHu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(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_nums_num),h4s_options_optionu_u_map(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27b),V_t)))))).
fof(ah4s_llists_LLENGTHu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_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_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_LTAKEu_u_EQu_u_SOMEu_u_CONS, axiom, ![TV_u_27a]: ![V_x, V_n, V_l]: (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_l))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_some(s(t_h4s_lists_list(TV_u_27a),V_x))) => ![V_h]: ?[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),h4s_llists_lcons(s(TV_u_27a,V_h),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),V_y))))).
fof(ah4s_llists_LTAKEu_u_NILu_u_EQu_u_SOME, axiom, ![TV_u_27a]: ![V_m, V_l]: (s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,V_m),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_some(s(t_h4s_lists_list(TV_u_27a),V_l))) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) & s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_llists_LTAKEu_u_SNOCu_u_LNTH, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_l, V_e]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(t_h4s_lists_list(TV_u_27a),V_l))),s(TV_u_27a,V_e))) = 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_append(s(t_h4s_lists_list(TV_u_27a),V_l),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_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))))) => ![V_uu_0]: (![V_n, V_ll, V_l]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_llists_lnth(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_none),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(t_h4s_lists_list(TV_u_27a),V_l))))) => ![V_n, V_ll]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_optionu_u_case(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_none),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll)))))))).
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_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_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_llists_LTAKE0u_c0, axiom, ![TV_u_27a]: ![V_ll]: 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_ll))) = 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_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_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_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_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_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_llists_LTAKEu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_n]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27b)),h4s_llists_ltake(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27b)),h4s_options_none)).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_llists_LNTHu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_n]: s(t_h4s_options_option(TV_u_27a),h4s_llists_lnth(s(t_h4s_nums_num,V_n),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_LNTH0u_c0, axiom, ![TV_u_27a]: ![V_ll]: s(t_h4s_options_option(TV_u_27a),h4s_llists_lnth(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),V_ll)))).
fof(ah4s_lists_APPENDu_u_11u_c0, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l3))) <=> s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),V_l3))).
fof(ah4s_llists_LNTHu_u_THMu_c2, axiom, ![TV_u_27c]: ![V_t, V_n, V_h]: s(t_h4s_options_option(TV_u_27c),h4s_llists_lnth(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_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(TV_u_27c),h4s_llists_lnth(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27c),V_t)))).
fof(ah4s_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(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_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = 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),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ch4s_llists_LFINITEu_u_fromList, conjecture, ![TV_u_27a]: ![V_l]: p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_fromlist(s(t_h4s_lists_list(TV_u_27a),V_l))))))).
