%   ORIGINAL: h4/lbtree/lbtree__case__thm_c1
% 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/lbtree/lbtree__case__def: !t f e. h4/lbtree/lbtree__case e f t = h4/bool/COND (t = h4/lbtree/Lf) e (f (h4/min/_40 (\a. ?t1 t2. t = h4/lbtree/Nd a t1 t2)) (h4/min/_40 (\t1. ?a t2. t = h4/lbtree/Nd a t1 t2)) (h4/min/_40 (\t2. ?a t1. t = h4/lbtree/Nd a t1 t2)))
% Assm: h4/lbtree/lbtree__case__thm_c0: !f e. h4/lbtree/lbtree__case e f h4/lbtree/Lf = e
% Assm: h4/lbtree/lbtree__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION h4/lbtree/is__lbtree rep
% Assm: h4/lbtree/Nd__def: !t2 t1 a. h4/lbtree/Nd a t1 t2 = h4/lbtree/lbtree__abs (h4/lbtree/Ndrep a (h4/lbtree/lbtree__rep t1) (h4/lbtree/lbtree__rep t2))
% Assm: h4/lbtree/lbtree__ue__Axiom: !f. h4/bool/_3F_21 (\g. !x. g x = h4/option/option__CASE (f x) h4/lbtree/Lf (\v. h4/pair/pair__CASE v (\b v2. h4/pair/pair__CASE v2 (\y z. h4/lbtree/Nd b (g y) (g z)))))
% Assm: h4/lbtree/lbtree__cases: !t. t = h4/lbtree/Lf \/ (?a t1 t2. t = h4/lbtree/Nd a t1 t2)
% Assm: h4/lbtree/Lf__NOT__Nd: !t2 t1 a. ~(h4/lbtree/Lf = h4/lbtree/Nd a t1 t2)
% Assm: h4/lbtree/Nd__11: !u2 u1 t2 t1 a2 a1. h4/lbtree/Nd a1 t1 u1 = h4/lbtree/Nd a2 t2 u2 <=> a1 = a2 /\ t1 = t2 /\ u1 = u2
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/lbtree/lbtree__absrep_c1: !r. h4/lbtree/is__lbtree r <=> h4/lbtree/lbtree__rep (h4/lbtree/lbtree__abs r) = r
% Assm: h4/lbtree/lbtree__absrep_c0: !a. h4/lbtree/lbtree__abs (h4/lbtree/lbtree__rep a) = a
% Assm: h4/lbtree/Lf__def: h4/lbtree/Lf = h4/lbtree/lbtree__abs h4/lbtree/Lfrep
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/lbtree/is__lbtree__def: !t. h4/lbtree/is__lbtree t <=> (?P. (!t0. P t0 ==> t0 = h4/lbtree/Lfrep \/ (?a t1 t2. P t1 /\ P t2 /\ t0 = h4/lbtree/Ndrep a t1 t2)) /\ P t)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/lbtree/Ndrep__def: !t2 t1 a. h4/lbtree/Ndrep a t1 t2 = (\l. h4/list/list__CASE l (h4/option/SOME a) (\v xs. h4/bool/COND v (t1 xs) (t2 xs)))
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/lbtree/Lfrep__def: h4/lbtree/Lfrep = (\l. h4/option/NONE)
% Assm: h4/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/bool/DISJ__IMP__THM: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/lbtree/path__follow__def_c1: !x t h g. h4/lbtree/path__follow g x (h4/list/CONS h t) = h4/option/option__CASE (g x) h4/option/NONE (\v. h4/pair/pair__CASE v (\a v2. h4/pair/pair__CASE v2 (\y z. h4/lbtree/path__follow g (h4/bool/COND h y z) t)))
% Assm: h4/lbtree/path__follow__def_c0: !x g. h4/lbtree/path__follow g x h4/list/NIL = h4/option/OPTION__MAP h4/pair/FST (g x)
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/pair/pair__CASES: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% 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/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/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/EXISTS__UNIQUE__THM: !P. h4/bool/_3F_21 (\x. P x) <=> (?x. P x) /\ (!x y. P x /\ P y ==> x = y)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/pair/ELIM__PEXISTS: !P. (?p. P (h4/pair/FST p) (h4/pair/SND p)) <=> (?p1 p2. P p1 p2)
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/option/option__Axiom: !f e. ?fn. fn h4/option/NONE = e /\ (!x. fn (h4/option/SOME x) = f x)
% Assm: h4/llist/LUNFOLD__EQ: !s ll f R. R s ll /\ (!s0 ll0. R s0 ll0 ==> f s0 = h4/option/NONE /\ ll0 = h4/llist/LNIL \/ (?s_27 x ll_27. f s0 = h4/option/SOME (h4/pair/_2C s_27 x) /\ h4/llist/LHD ll0 = h4/option/SOME x /\ h4/llist/LTL ll0 = h4/option/SOME ll_27 /\ R s_27 ll_27)) ==> h4/llist/LUNFOLD f s = ll
% Assm: h4/pred__set/ITSET__INSERT: !s. h4/pred__set/FINITE s ==> (!f x b. h4/pred__set/ITSET f (h4/pred__set/INSERT x s) b = h4/pred__set/ITSET f (h4/pred__set/REST (h4/pred__set/INSERT x s)) (f (h4/pred__set/CHOICE (h4/pred__set/INSERT x s)) b))
% Assm: h4/llist/LUNFOLD__THM_c0: !x f. f x = h4/option/NONE ==> h4/llist/LUNFOLD f x = h4/llist/LNIL
% Assm: h4/llist/LUNFOLD__THM_c1: !x v2 v1 f. f x = h4/option/SOME (h4/pair/_2C v1 v2) ==> h4/llist/LUNFOLD f x = h4/llist/LCONS v2 (h4/llist/LUNFOLD f v1)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/llist/LHD__EQ__NONE_c1: !ll. h4/option/NONE = h4/llist/LHD ll <=> ll = h4/llist/LNIL
% Assm: h4/llist/LLIST__BISIMULATION0: !ll2 ll1. ll1 = ll2 <=> (?R. R ll1 ll2 /\ (!ll3 ll4. R ll3 ll4 ==> ll3 = h4/llist/LNIL /\ ll4 = h4/llist/LNIL \/ (?h t1 t2. ll3 = h4/llist/LCONS h t1 /\ ll4 = h4/llist/LCONS h t2 /\ R t1 t2)))
% 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/LHD__THM_c1: !t h. h4/llist/LHD (h4/llist/LCONS h t) = h4/option/SOME h
% Assm: h4/llist/LHD__THM_c0: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/LTL__THM_c0: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/llist__CASES: !l. l = h4/llist/LNIL \/ (?h t. l = h4/llist/LCONS h t)
% Assm: h4/llist/LTL__THM_c1: !t h. h4/llist/LTL (h4/llist/LCONS h t) = h4/option/SOME t
% Assm: h4/llist/LCONS__NOT__NIL_c0: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/pred__set/ITSET__curried__def: !x1 x f. h4/pred__set/ITSET f x x1 = h4/pred__set/ITSET__tupled f (h4/pair/_2C x x1)
% Assm: h4/pred__set/ITSET__tupled__primitive__def: !f. h4/pred__set/ITSET__tupled f = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!b s. h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> R (h4/pair/_2C (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b)) (h4/pair/_2C s b)))) (\ITSET__tupled a. h4/pair/pair__CASE a (\s b. h4/combin/I (h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND (s = h4/pred__set/EMPTY) b (ITSET__tupled (h4/pair/_2C (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b)))) h4/bool/ARB)))
% Assm: h4/pred__set/REST__PSUBSET: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/PSUBSET (h4/pred__set/REST s) s
% Assm: h4/pred__set/NOT__INSERT__EMPTY: !x s. ~(h4/pred__set/INSERT x s = h4/pred__set/EMPTY)
% Assm: h4/pred__set/FINITE__INSERT: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pred__set/CARD__PSUBSET: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/PSUBSET t s ==> h4/prim__rec/_3C (h4/pred__set/CARD t) (h4/pred__set/CARD s))
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/prim__rec/measure__def: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Goal: !t2 t1 f e a. h4/lbtree/lbtree__case e f (h4/lbtree/Nd a t1 t2) = f a t1 t2
%   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_lbtrees_lbtreeu_u_caseu_u_def]: !_2. (!t t2. happ (happ _2 t) t2 <=> (?a t1. t = h4/lbtree/Nd a t1 t2)) ==> (!_1. (!t t1. happ (happ _1 t) t1 <=> (?a t2. t = h4/lbtree/Nd a t1 t2)) ==> (!_0. (!t a. happ (happ _0 t) a <=> (?t1 t2. t = h4/lbtree/Nd a t1 t2)) ==> (!t f e. ?v. (v <=> t = h4/lbtree/Lf) /\ h4/lbtree/lbtree__case e f t = h4/bool/COND v e (happ (happ (happ f (h4/min/_40 (happ _0 t))) (h4/min/_40 (happ _1 t))) (h4/min/_40 (happ _2 t))))))
% Assm [h4s_lbtrees_lbtreeu_u_caseu_u_thmu_c0]: !f e. h4/lbtree/lbtree__case e f h4/lbtree/Lf = e
% Assm [h4s_lbtrees_lbtreeu_u_TYu_u_DEF]: ?rep. h4/bool/TYPE__DEFINITION h4/lbtree/is__lbtree rep
% Assm [h4s_lbtrees_Ndu_u_def]: !t2 t1 a. h4/lbtree/Nd a t1 t2 = h4/lbtree/lbtree__abs (h4/lbtree/Ndrep a (h4/lbtree/lbtree__rep t1) (h4/lbtree/lbtree__rep t2))
% Assm [h4s_lbtrees_lbtreeu_u_ueu_u_Axiom]: !_5. (!b y g z. happ (happ (happ (happ _5 b) y) g) z = h4/lbtree/Nd b (happ g y) (happ g z)) ==> (!_4. (!b g y. happ (happ (happ _4 b) g) y = happ (happ (happ _5 b) y) g) ==> (!_3. (!b g v2. happ (happ (happ _3 b) g) v2 = h4/pair/pair__CASE v2 (happ (happ _4 b) g)) ==> (!_2. (!g b. happ (happ _2 g) b = happ (happ _3 b) g) ==> (!_1. (!g v. happ (happ _1 g) v = h4/pair/pair__CASE v (happ _2 g)) ==> (!_0. (!f g. happ (happ _0 f) g <=> (!x. happ g x = h4/option/option__CASE (happ f x) h4/lbtree/Lf (happ _1 g))) ==> (!f. h4/bool/_3F_21 (happ _0 f)))))))
% Assm [h4s_lbtrees_lbtreeu_u_cases]: !t. t = h4/lbtree/Lf \/ (?a t1 t2. t = h4/lbtree/Nd a t1 t2)
% Assm [h4s_lbtrees_Lfu_u_NOTu_u_Nd]: !t2 t1 a. ~(h4/lbtree/Lf = h4/lbtree/Nd a t1 t2)
% Assm [h4s_lbtrees_Ndu_u_11]: !u2 u1 t2 t1 a2 a1. h4/lbtree/Nd a1 t1 u1 = h4/lbtree/Nd a2 t2 u2 <=> a1 = a2 /\ t1 = t2 /\ u1 = u2
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_lbtrees_lbtreeu_u_absrepu_c1]: !r. happ h4/lbtree/is__lbtree r <=> h4/lbtree/lbtree__rep (h4/lbtree/lbtree__abs r) = r
% Assm [h4s_lbtrees_lbtreeu_u_absrepu_c0]: !a. h4/lbtree/lbtree__abs (h4/lbtree/lbtree__rep a) = a
% Assm [h4s_lbtrees_Lfu_u_def]: h4/lbtree/Lf = h4/lbtree/lbtree__abs h4/lbtree/Lfrep
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_lbtrees_isu_u_lbtreeu_u_def]: !t. happ h4/lbtree/is__lbtree t <=> (?P. (!t0. happ P t0 ==> t0 = h4/lbtree/Lfrep \/ (?a t1 t2. happ P t1 /\ happ P t2 /\ t0 = h4/lbtree/Ndrep a t1 t2)) /\ happ P t)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_lbtrees_Ndrepu_u_def]: !_1. (!v t1 t2 xs. happ (happ (happ (happ _1 v) t1) t2) xs = h4/bool/COND v (happ t1 xs) (happ t2 xs)) ==> (!_0. (!t1 t2 v. happ (happ (happ _0 t1) t2) v = happ (happ (happ _1 v) t1) t2) ==> (!t2 t1 a x. happ (h4/lbtree/Ndrep a t1 t2) x = h4/list/list__CASE x (h4/option/SOME a) (happ (happ _0 t1) t2)))
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lbtrees_Lfrepu_u_def]: !x. happ h4/lbtree/Lfrep x = h4/option/NONE
% 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_DISJu_u_IMPu_u_THM]: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_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_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_lbtrees_pathu_u_followu_u_defu_c1]: !_4. (!g h y t z. happ (happ (happ (happ (happ _4 g) h) y) t) z = h4/lbtree/path__follow g (h4/bool/COND h y z) t) ==> (!_3. (!g h t y. happ (happ (happ (happ _3 g) h) t) y = happ (happ (happ (happ _4 g) h) y) t) ==> (!_2. (!g h t v2. happ (happ (happ (happ _2 g) h) t) v2 = h4/pair/pair__CASE v2 (happ (happ (happ _3 g) h) t)) ==> (!_1. (!g h t a. happ (happ (happ (happ _1 g) h) t) a = happ (happ (happ _2 g) h) t) ==> (!_0. (!g h t v. happ (happ (happ (happ _0 g) h) t) v = h4/pair/pair__CASE v (happ (happ (happ _1 g) h) t)) ==> (!x t h g. h4/lbtree/path__follow g x (h4/list/CONS h t) = h4/option/option__CASE (happ g x) h4/option/NONE (happ (happ (happ _0 g) h) t))))))
% Assm [h4s_lbtrees_pathu_u_followu_u_defu_c0]: !x g. h4/lbtree/path__follow g x h4/list/NIL = h4/option/OPTION__MAP h4/pair/FST (happ g x)
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_pairs_pairu_u_CASES]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% 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_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_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_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_THM]: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. h4/bool/_3F_21 (happ _0 P) <=> (?x. happ P x) /\ (!x y. happ P x /\ happ P y ==> x = y))
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_pairs_ELIMu_u_PEXISTS]: !P. (?p. happ (happ P (happ h4/pair/FST p)) (h4/pair/SND p)) <=> (?p1 p2. happ (happ P p1) p2)
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_options_optionu_u_Axiom]: !f e. ?fn. happ fn h4/option/NONE = e /\ (!x. happ fn (h4/option/SOME x) = happ f x)
% Assm [h4s_llists_LUNFOLDu_u_EQ]: !s ll f R. happ (happ R s) ll /\ (!s0 ll0. happ (happ R s0) ll0 ==> happ f s0 = h4/option/NONE /\ ll0 = h4/llist/LNIL \/ (?s_27 x ll_27. happ f s0 = h4/option/SOME (h4/pair/_2C s_27 x) /\ h4/llist/LHD ll0 = h4/option/SOME x /\ h4/llist/LTL ll0 = h4/option/SOME ll_27 /\ happ (happ R s_27) ll_27)) ==> h4/llist/LUNFOLD f s = ll
% Assm [h4s_predu_u_sets_ITSETu_u_INSERT]: !s. h4/pred__set/FINITE s ==> (!f x b. h4/pred__set/ITSET f (h4/pred__set/INSERT x s) b = h4/pred__set/ITSET f (h4/pred__set/REST (h4/pred__set/INSERT x s)) (happ (happ f (h4/pred__set/CHOICE (h4/pred__set/INSERT x s))) b))
% Assm [h4s_llists_LUNFOLDu_u_THMu_c0]: !x f. happ f x = h4/option/NONE ==> h4/llist/LUNFOLD f x = h4/llist/LNIL
% Assm [h4s_llists_LUNFOLDu_u_THMu_c1]: !x v2 v1 f. happ f x = h4/option/SOME (h4/pair/_2C v1 v2) ==> h4/llist/LUNFOLD f x = h4/llist/LCONS v2 (h4/llist/LUNFOLD f v1)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_llists_LHDu_u_EQu_u_NONEu_c1]: !ll. h4/option/NONE = h4/llist/LHD ll <=> ll = h4/llist/LNIL
% Assm [h4s_llists_LLISTu_u_BISIMULATION0]: !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 \/ (?h t1 t2. ll3 = h4/llist/LCONS h t1 /\ ll4 = h4/llist/LCONS h t2 /\ happ (happ R t1) t2)))
% 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_LHDu_u_THMu_c1]: !t h. h4/llist/LHD (h4/llist/LCONS h t) = h4/option/SOME h
% Assm [h4s_llists_LHDu_u_THMu_c0]: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_LTLu_u_THMu_c0]: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_llistu_u_CASES]: !l. l = h4/llist/LNIL \/ (?h t. l = h4/llist/LCONS h t)
% Assm [h4s_llists_LTLu_u_THMu_c1]: !t h. h4/llist/LTL (h4/llist/LCONS h t) = h4/option/SOME t
% Assm [h4s_llists_LCONSu_u_NOTu_u_NILu_c0]: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (h4/option/SOME x) = x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_predu_u_sets_ITSETu_u_curriedu_u_def]: !x1 x f. h4/pred__set/ITSET f x x1 = happ (h4/pred__set/ITSET__tupled f) (h4/pair/_2C x x1)
% Assm [h4s_predu_u_sets_ITSETu_u_tupledu_u_primitiveu_u_def]: !_4. (!ITSET__tupled f s b. ?v. (v <=> s = h4/pred__set/EMPTY) /\ happ (happ (happ (happ _4 ITSET__tupled) f) s) b = h4/combin/I (h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND v b (happ ITSET__tupled (h4/pair/_2C (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b)))) h4/bool/ARB)) ==> (!_3. (!ITSET__tupled f s. happ (happ (happ _3 ITSET__tupled) f) s = happ (happ (happ _4 ITSET__tupled) f) s) ==> (!_2. (!ITSET__tupled f a. happ (happ (happ _2 ITSET__tupled) f) a = h4/pair/pair__CASE a (happ (happ _3 ITSET__tupled) f)) ==> (!_1. (!f ITSET__tupled. happ (happ _1 f) ITSET__tupled = happ (happ _2 ITSET__tupled) f) ==> (!_0. (!f R. happ (happ _0 f) R <=> h4/relation/WF R /\ (!b s. h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> happ (happ R (h4/pair/_2C (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b))) (h4/pair/_2C s b))) ==> (!f. h4/pred__set/ITSET__tupled f = h4/relation/WFREC (h4/min/_40 (happ _0 f)) (happ _1 f))))))
% Assm [h4s_predu_u_sets_RESTu_u_PSUBSET]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/PSUBSET (h4/pred__set/REST s) s
% Assm [h4s_predu_u_sets_NOTu_u_INSERTu_u_EMPTY]: !x s. ~(h4/pred__set/INSERT x s = h4/pred__set/EMPTY)
% Assm [h4s_predu_u_sets_FINITEu_u_INSERT]: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_predu_u_sets_CARDu_u_PSUBSET]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/PSUBSET t s ==> happ (happ h4/prim__rec/_3C (h4/pred__set/CARD t)) (h4/pred__set/CARD s))
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (happ h4/prim__rec/measure m)
% Assm [h4s_primu_u_recs_measureu_u_def]: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (happ (h4/relation/inv__image R) f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_combins_ou_u_DEF]: !g f x. h4/combin/o f g x = happ f (happ g x)
% Goal: !t2 t1 f e a. h4/lbtree/lbtree__case e f (h4/lbtree/Nd a t1 t2) = happ (happ (happ f a) t1) t2
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1287565,TV_Q1287561]: ![V_f, V_g]: (![V_x]: s(TV_Q1287561,happ(s(t_fun(TV_Q1287565,TV_Q1287561),V_f),s(TV_Q1287565,V_x))) = s(TV_Q1287561,happ(s(t_fun(TV_Q1287565,TV_Q1287561),V_g),s(TV_Q1287565,V_x))) => s(t_fun(TV_Q1287565,TV_Q1287561),V_f) = s(t_fun(TV_Q1287565,TV_Q1287561),V_g))).
fof(ah4s_lbtrees_lbtreeu_u_caseu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_t, V_t2]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool)),V_uu_2),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))) <=> ?[V_a, V_t1]: s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))) => ![V_uu_1]: (![V_t, V_t1]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool)),V_uu_1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1)))) <=> ?[V_a, V_t2]: s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))) => ![V_uu_0]: (![V_t, V_a]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))),s(TV_u_27b,V_a)))) <=> ?[V_t1, V_t2]: s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))) => ![V_t, V_f, V_e]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_lf)) & s(TV_u_27a,h4s_lbtrees_lbtreeu_u_case(s(TV_u_27a,V_e),s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,V_e),s(TV_u_27a,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(TV_u_27b,h4s_mins_u_40(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(TV_u_27b,t_bool)),V_uu_0),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))))))),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_mins_u_40(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool)),V_uu_1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t))))))),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_mins_u_40(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_bool)),V_uu_2),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t)))))))))))))).
fof(ah4s_lbtrees_lbtreeu_u_caseu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: s(TV_u_27a,h4s_lbtrees_lbtreeu_u_case(s(TV_u_27a,V_e),s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_lf))) = s(TV_u_27a,V_e)).
fof(ah4s_lbtrees_lbtreeu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),h4s_lbtrees_isu_u_lbtree),s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),V_rep))))).
fof(ah4s_lbtrees_Ndu_u_def, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_a]: s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_ndrep(s(TV_u_27a,V_a),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))))))).
fof(ah4s_lbtrees_lbtreeu_u_ueu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_5]: (![V_b, V_y, V_g, V_z]: s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))))),V_uu_5),s(TV_u_27b,V_b))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(TV_u_27a,V_z))) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g),s(TV_u_27a,V_y))),s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g),s(TV_u_27a,V_z))))) => ![V_uu_4]: (![V_b, V_g, V_y]: s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))))),V_uu_4),s(TV_u_27b,V_b))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(TV_u_27a,V_y))) = s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))))),V_uu_5),s(TV_u_27b,V_b))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))) => ![V_uu_3]: (![V_b, V_g, V_v2]: s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)))),V_uu_3),s(TV_u_27b,V_b))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_v2))) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_v2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))))),V_uu_4),s(TV_u_27b,V_b))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))))) => ![V_uu_2]: (![V_g, V_b]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(TV_u_27b,V_b))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)))),V_uu_3),s(TV_u_27b,V_b))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))) => ![V_uu_1]: (![V_g, V_v]: s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),t_h4s_lbtrees_lbtree(TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),V_v))) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),V_v),s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))))) => ![V_uu_0]: (![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),V_f))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g)))) <=> ![V_x]: s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g),s(TV_u_27a,V_x))) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a))),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),V_f),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_lf),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),t_h4s_lbtrees_lbtree(TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g)))))) => ![V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),V_f))))))))))))).
fof(ah4s_lbtrees_lbtreeu_u_cases, axiom, ![TV_u_27a]: ![V_t]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) | ?[V_a, V_t1, V_t2]: s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))).
fof(ah4s_lbtrees_Lfu_u_NOTu_u_Nd, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_a]: ~ (s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))).
fof(ah4s_lbtrees_Ndu_u_11, axiom, ![TV_u_27a]: ![V_u2, V_u1, V_t2, V_t1, V_a2, V_a1]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u1))) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a2),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u2))) <=> (s(TV_u_27a,V_a1) = s(TV_u_27a,V_a2) & (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2) & s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u1) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u2))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_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_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_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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_lbtrees_lbtreeu_u_absrepu_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),h4s_lbtrees_isu_u_lbtree),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_r)))) <=> s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_r))))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_r))).
fof(ah4s_lbtrees_lbtreeu_u_absrepu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a))))) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a)).
fof(ah4s_lbtrees_Lfu_u_def, axiom, ![TV_u_27a]: s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lfrep)))).
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_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_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_lbtrees_isu_u_lbtreeu_u_def, axiom, ![TV_u_27a]: ![V_t]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),h4s_lbtrees_isu_u_lbtree),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t)))) <=> ?[V_P]: (![V_t0]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),V_P),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t0)))) => (s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t0) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lfrep) | ?[V_a, V_t1, V_t2]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),V_P),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1)))) & (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),V_P),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2)))) & s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t0) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_ndrep(s(TV_u_27a,V_a),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))))))) & p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),V_P),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),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_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,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_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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_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_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_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
fof(ah4s_sats_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_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, 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_A)) | p(s(t_bool,V_C)))))).
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_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_ORu_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_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, 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_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_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_lbtrees_Ndrepu_u_def, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_v, V_t1, V_t2, V_xs]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_1),s(t_bool,V_v))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_h4s_lists_list(t_bool),V_xs))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1),s(t_h4s_lists_list(t_bool),V_xs))),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2),s(t_h4s_lists_list(t_bool),V_xs))))) => ![V_uu_0]: (![V_t1, V_t2, V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_bool,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_1),s(t_bool,V_v))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))) => ![V_t2, V_t1, V_a, V_x]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_ndrep(s(TV_u_27a,V_a),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_h4s_lists_list(t_bool),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_a))),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2)))))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
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_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(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,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
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,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lbtrees_Lfrepu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),h4s_lbtrees_lfrep),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
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_DISJu_u_IMPu_u_THM, axiom, ![V_R, V_Q, V_P]: (((p(s(t_bool,V_P)) | p(s(t_bool,V_Q))) => p(s(t_bool,V_R))) <=> ((p(s(t_bool,V_P)) => p(s(t_bool,V_R))) & (p(s(t_bool,V_Q)) => p(s(t_bool,V_R)))))).
fof(ah4s_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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
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_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_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_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
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_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_lbtrees_pathu_u_followu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_4]: (![V_g, V_h, V_y, V_t, V_z]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))))),V_uu_4),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(TV_u_27b,V_y))),s(t_h4s_lists_list(t_bool),V_t))),s(TV_u_27b,V_z))) = s(t_h4s_options_option(TV_u_27a),h4s_lbtrees_pathu_u_follow(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,h4s_bools_cond(s(t_bool,V_h),s(TV_u_27b,V_y),s(TV_u_27b,V_z))),s(t_h4s_lists_list(t_bool),V_t))) => ![V_uu_3]: (![V_g, V_h, V_t, V_y]: s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))))),V_uu_3),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))),s(TV_u_27b,V_y))) = s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))))),V_uu_4),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(TV_u_27b,V_y))),s(t_h4s_lists_list(t_bool),V_t))) => ![V_uu_2]: (![V_g, V_h, V_t, V_v2]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))))),V_uu_2),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),V_v2))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),V_v2),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))))),V_uu_3),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))))) => ![V_uu_1]: (![V_g, V_h, V_t, V_a]: s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))))),V_uu_1),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))),s(TV_u_27a,V_a))) = s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))))),V_uu_2),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))) => ![V_uu_0]: (![V_g, V_h, V_t, V_v]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),V_v))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))))),V_uu_1),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))))) => ![V_x, V_t, V_h, V_g]: s(t_h4s_options_option(TV_u_27a),h4s_lbtrees_pathu_u_follow(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,V_x),s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,V_h),s(t_h4s_lists_list(t_bool),V_t))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b))),happ(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))))))))))).
fof(ah4s_lbtrees_pathu_u_followu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_g]: s(t_h4s_options_option(TV_u_27a),h4s_lbtrees_pathu_u_follow(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,V_x),s(t_h4s_lists_list(t_bool),h4s_lists_nil))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),TV_u_27a),h4s_pairs_fst),s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b))),happ(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,V_x)))))).
fof(ah4s_lists_listu_u_nchotomy, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) | ?[V_h, V_t]: s(t_h4s_lists_list(TV_u_27a),V_l) = s(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))))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(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)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_pairs_pairu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_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_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_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_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_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_EXISTSu_u_UNIQUEu_u_THM, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))) => ![V_P]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),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, V_y]: ((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_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
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_pairs_ELIMu_u_PEXISTS, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (?[V_p]: 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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))) <=> ?[V_p1, V_p2]: 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_p1))),s(TV_u_27b,V_p2)))))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_options_optionu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(TV_u_27b,V_e) & ![V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_some(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_LUNFOLDu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_ll, V_f, V_R]: ((p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_llists_llist(TV_u_27b),t_bool)),V_R),s(TV_u_27a,V_s))),s(t_h4s_llists_llist(TV_u_27b),V_ll)))) & ![V_s0, V_ll0]: (p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_llists_llist(TV_u_27b),t_bool)),V_R),s(TV_u_27a,V_s0))),s(t_h4s_llists_llist(TV_u_27b),V_ll0)))) => ((s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_options_none) & s(t_h4s_llists_llist(TV_u_27b),V_ll0) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil)) | ?[V_su_27, V_x, V_llu_27]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_su_27),s(TV_u_27b,V_x))))) & (s(t_h4s_options_option(TV_u_27b),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27b),V_ll0))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_x))) & (s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27b),V_ll0))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_options_some(s(t_h4s_llists_llist(TV_u_27b),V_llu_27))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_llists_llist(TV_u_27b),t_bool)),V_R),s(TV_u_27a,V_su_27))),s(t_h4s_llists_llist(TV_u_27b),V_llu_27)))))))))) => s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_s))) = s(t_h4s_llists_llist(TV_u_27b),V_ll))).
fof(ah4s_predu_u_sets_ITSETu_u_INSERT, axiom, ![TV_u_27a,TV_u_27b]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f, V_x, V_b]: s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,V_b))) = s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))),s(TV_u_27b,V_b))))))).
fof(ah4s_llists_LUNFOLDu_u_THMu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_options_none) => s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil))).
fof(ah4s_llists_LUNFOLDu_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v2, V_v1, V_f]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_v1),s(TV_u_27b,V_v2))))) => s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_v2),s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_v1))))))).
fof(ah4s_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_llists_LHDu_u_EQu_u_NONEu_c1, axiom, ![TV_u_27a]: ![V_ll]: (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),V_ll))) <=> s(t_h4s_llists_llist(TV_u_27a),V_ll) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))).
fof(ah4s_llists_LLISTu_u_BISIMULATION0, 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)) | ?[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_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_LHDu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(TV_u_27b),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_h),s(t_h4s_llists_llist(TV_u_27b),V_t))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_h)))).
fof(ah4s_llists_LHDu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_llists_LTLu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_options_none)).
fof(ah4s_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_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_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_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_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t)).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_predu_u_sets_ITSETu_u_curriedu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x1, V_x, V_f]: s(TV_u_27b,h4s_predu_u_sets_itset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27b,V_x1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),h4s_predu_u_sets_itsetu_u_tupled(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27b,V_x1)))))).
fof(ah4s_predu_u_sets_ITSETu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_4]: (![V_ITSETu_u_tupled, V_f, V_s, V_b]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,V_b))) = s(TV_u_27b,h4s_combins_i(s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))))),s(TV_u_27b,h4s_bools_arb)))))) => ![V_uu_3]: (![V_ITSETu_u_tupled, V_f, V_s]: s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))) => ![V_uu_2]: (![V_ITSETu_u_tupled, V_f, V_a]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),V_a))) = s(TV_u_27b,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),V_a),s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))) => ![V_uu_1]: (![V_f, V_ITSETu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))) => ![V_uu_0]: (![V_f, V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R)))) & ![V_b, V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27b,V_b))))))))) => ![V_f]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),h4s_predu_u_sets_itsetu_u_tupled(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))))))))).
fof(ah4s_predu_u_sets_RESTu_u_PSUBSET, axiom, ![TV_u_27a]: ![V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_NOTu_u_INSERTu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x, V_s]: ~ (s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
fof(ah4s_predu_u_sets_FINITEu_u_INSERT, axiom, ![TV_u_27a]: ![V_x, V_s]: s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_predu_u_sets_CARDu_u_PSUBSET, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_t]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_t))))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_primu_u_recs_WFu_u_measure, axiom, ![TV_u_27a]: ![V_m]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure),s(t_fun(TV_u_27a,t_h4s_nums_num),V_m))))))).
fof(ah4s_primu_u_recs_measureu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure) = s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),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_y))))) => 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_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
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_LEFTu_u_FORALLu_u_ORu_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_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ch4s_lbtrees_lbtreeu_u_caseu_u_thmu_c1, conjecture, ![TV_u_27a,TV_u_27b]: ![V_t2, V_t1, V_f, V_e, V_a]: s(TV_u_27a,h4s_lbtrees_lbtreeu_u_case(s(TV_u_27a,V_e),s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(TV_u_27b,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))).
