%   ORIGINAL: h4/lbtree/lbtree__case__thm_c0
% 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/Lf__def: h4/lbtree/Lf = h4/lbtree/lbtree__abs h4/lbtree/Lfrep
% 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/lbtree__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION h4/lbtree/is__lbtree rep
% 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/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__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/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/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/IMP__CLAUSES_c1: !t. t ==> T <=> 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/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/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% 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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% 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/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% 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/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/lbtree/Lfrep__def: h4/lbtree/Lfrep = (\l. h4/option/NONE)
% Assm: h4/bool/DISJ__IMP__THM: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm: h4/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% 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/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% 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/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% 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/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/pair/pair__CASES: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% 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/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% 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/AND__CLAUSES_c4: !t. 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/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/relation/RESTRICT__DEF: !x f R. h4/relation/RESTRICT f R x = (\y. h4/bool/COND (R y x) (f y) h4/bool/ARB)
% Assm: h4/pair/UNCURRY__CONG: !f_27 f M_27 M. M = M_27 /\ (!x y. M_27 = h4/pair/_2C x y ==> f x y = f_27 x y) ==> h4/pair/UNCURRY f M = h4/pair/UNCURRY f_27 M_27
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/relation/RSUBSET0: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. R1 x y ==> R2 x y)
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/set__relation/rel__to__reln__def: !R. h4/set__relation/rel__to__reln R = h4/pred__set/GSPEC (h4/pair/UNCURRY (\x y. h4/pair/_2C (h4/pair/_2C x y) (R x y)))
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/pair/prod__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\p. ?x y. p = (\a b. a = x /\ b = y)) rep
% Assm: h4/pair/ABS__REP__prod_c1: !r. (\p. ?x y. p = (\a b. a = x /\ b = y)) r <=> h4/pair/REP__prod (h4/pair/ABS__prod r) = r
% Assm: h4/bool/itself__Axiom: !e. ?f. f h4/bool/the__value = e
% Assm: h4/relation/TFL__INDUCTIVE__INVARIANT__ON__WFREC: !x f R P M D. f = h4/relation/WFREC R M /\ h4/relation/WF R /\ h4/relation/INDUCTIVE__INVARIANT__ON R D P M /\ D x ==> P x (f x)
% Assm: h4/relation/TC__lifts__equalities: !f R. (!x y. R x y ==> f x = f y) ==> (!x y. h4/relation/TC R x y ==> f x = f y)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/relation/INDUCTIVE__INVARIANT__ON__WFREC: !x R P M D. h4/relation/WF R /\ h4/relation/INDUCTIVE__INVARIANT__ON R D P M /\ D x ==> P x (h4/relation/WFREC R M x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/sum/OUTR0: !x. h4/sum/OUTR (h4/sum/INR x) = x
% Assm: h4/relation/TC__INDUCT: !R P. (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ P y z ==> P x z) ==> (!u v. h4/relation/TC R u v ==> P u v)
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/sum/sum__axiom: !g f. h4/bool/_3F_21 (\h. h4/combin/o h h4/sum/INL = f /\ h4/combin/o h h4/sum/INR = g)
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/llist/llist__Axiom: !f. ?g. (!x. h4/llist/LHD (g x) = h4/option/OPTION__MAP h4/pair/SND (f x)) /\ (!x. h4/llist/LTL (g x) = h4/option/OPTION__MAP (h4/combin/o g h4/pair/FST) (f x))
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/llist/llist__ue__Axiom: !f. h4/bool/_3F_21 (\g. (!x. h4/llist/LHD (g x) = h4/option/OPTION__MAP h4/pair/SND (f x)) /\ (!x. h4/llist/LTL (g x) = h4/option/OPTION__MAP (h4/combin/o g h4/pair/FST) (f x)))
% Assm: h4/option/option__CLAUSES_c14: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/relation/RDOM__DELETE__DEF: !x v u R. h4/relation/RDOM__DELETE R x u v <=> R u v /\ ~(u = x)
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/ind__type/CONSTR__IND: !P. P h4/ind__type/BOTTOM /\ (!c i r. (!n. P (r n)) ==> P (h4/ind__type/CONSTR c i r)) ==> (!x. P x)
% Assm: h4/ind__type/CONSTR__INJ: !r2 r1 i2 i1 c2 c1. h4/ind__type/CONSTR c1 i1 r1 = h4/ind__type/CONSTR c2 i2 r2 <=> c1 = c2 /\ i1 = i2 /\ r1 = r2
% Assm: h4/ind__type/CONSTR__BOT: !r i c. ~(h4/ind__type/CONSTR c i r = h4/ind__type/BOTTOM)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Goal: !f e. h4/lbtree/lbtree__case e f h4/lbtree/Lf = e
%   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_Lfu_u_def]: h4/lbtree/Lf = h4/lbtree/lbtree__abs h4/lbtree/Lfrep
% 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_lbtreeu_u_TYu_u_DEF]: ?rep. h4/bool/TYPE__DEFINITION h4/lbtree/is__lbtree rep
% 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_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_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_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_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_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% 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_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% 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_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% 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_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_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% 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_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_lbtrees_Lfrepu_u_def]: !x. happ h4/lbtree/Lfrep x = h4/option/NONE
% Assm [h4s_bools_DISJu_u_IMPu_u_THM]: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. happ P x ==> Q) <=> (?x. happ P x) ==> Q
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% 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_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% 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_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% 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_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_pairs_pairu_u_CASES]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% 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_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_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% 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_ANDu_u_CLAUSESu_c4]: !t. 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_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> h4/relation/RESTRICT f R z y = happ f y
% Assm [h4s_relations_RESTRICTu_u_DEF]: !x f R x'. h4/relation/RESTRICT f R x x' = h4/bool/COND (happ (happ R x') x) (happ f x') h4/bool/ARB
% Assm [h4s_pairs_UNCURRYu_u_CONG]: !f_27 f M_27 M. M = M_27 /\ (!x y. M_27 = h4/pair/_2C x y ==> happ (happ f x) y = happ (happ f_27 x) y) ==> happ (h4/pair/UNCURRY f) M = happ (h4/pair/UNCURRY f_27) M_27
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_relations_RSUBSET0]: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 x) y)
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_setu_u_relations_relu_u_tou_u_relnu_u_def]: !_1. (!R x y. happ (happ (happ _1 R) x) y = h4/pair/_2C (h4/pair/_2C x y) (happ (happ R x) y)) ==> (!_0. (!R x. happ (happ _0 R) x = happ (happ _1 R) x) ==> (!R. h4/set__relation/rel__to__reln R = h4/pred__set/GSPEC (h4/pair/UNCURRY (happ _0 R))))
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_pairs_produ_u_TYu_u_DEF]: !_0. (!p. happ _0 p <=> (?x y. !x' x. happ (happ p x') x <=> x' = x /\ x = y)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_pairs_ABSu_u_REPu_u_produ_c1]: !r. (?x y. !x' x. happ (happ r x') x <=> x' = x /\ x = y) <=> h4/pair/REP__prod (h4/pair/ABS__prod r) = r
% Assm [h4s_bools_itselfu_u_Axiom]: !e. ?f. happ f h4/bool/the__value = e
% Assm [h4s_relations_TFLu_u_INDUCTIVEu_u_INVARIANTu_u_ONu_u_WFREC]: !x f R P M D. f = h4/relation/WFREC R M /\ h4/relation/WF R /\ h4/relation/INDUCTIVE__INVARIANT__ON R D P M /\ happ D x ==> happ (happ P x) (happ f x)
% Assm [h4s_relations_TCu_u_liftsu_u_equalities]: !f R. (!x y. happ (happ R x) y ==> happ f x = happ f y) ==> (!x y. h4/relation/TC R x y ==> happ f x = happ f y)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_relations_INDUCTIVEu_u_INVARIANTu_u_ONu_u_WFREC]: !x R P M D. h4/relation/WF R /\ h4/relation/INDUCTIVE__INVARIANT__ON R D P M /\ happ D x ==> happ (happ P x) (happ (h4/relation/WFREC R M) x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_sums_OUTR0]: !x. h4/sum/OUTR (happ h4/sum/INR x) = x
% Assm [h4s_relations_TCu_u_INDUCT]: !R P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ P x) y /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!u v. h4/relation/TC R u v ==> happ (happ P u) v)
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_sums_sumu_u_axiom]: !_0. (!f g h. happ (happ (happ _0 f) g) h <=> h4/combin/o h h4/sum/INL = f /\ h4/combin/o h h4/sum/INR = g) ==> (!g f. h4/bool/_3F_21 (happ (happ _0 f) g))
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_llists_llistu_u_Axiom]: !f. ?g. (!x. h4/llist/LHD (happ g x) = h4/option/OPTION__MAP h4/pair/SND (happ f x)) /\ (!x. h4/llist/LTL (happ g x) = h4/option/OPTION__MAP (h4/combin/o g h4/pair/FST) (happ f x))
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_llists_llistu_u_ueu_u_Axiom]: !_0. (!f g. happ (happ _0 f) g <=> (!x. h4/llist/LHD (happ g x) = h4/option/OPTION__MAP h4/pair/SND (happ f x)) /\ (!x. h4/llist/LTL (happ g x) = h4/option/OPTION__MAP (h4/combin/o g h4/pair/FST) (happ f x))) ==> (!f. h4/bool/_3F_21 (happ _0 f))
% Assm [h4s_options_optionu_u_CLAUSESu_c14]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_relations_RDOMu_u_DELETEu_u_DEF]: !x v u R. h4/relation/RDOM__DELETE R x u v <=> happ (happ R u) v /\ ~(u = x)
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_indu_u_types_CONSTRu_u_IND]: !P. happ P h4/ind__type/BOTTOM /\ (!c i r. (!n. happ P (happ r n)) ==> happ P (h4/ind__type/CONSTR c i r)) ==> (!x. happ P x)
% Assm [h4s_indu_u_types_CONSTRu_u_INJ]: !r2 r1 i2 i1 c2 c1. h4/ind__type/CONSTR c1 i1 r1 = h4/ind__type/CONSTR c2 i2 r2 <=> c1 = c2 /\ i1 = i2 /\ r1 = r2
% Assm [h4s_indu_u_types_CONSTRu_u_BOT]: !r i c. ~(h4/ind__type/CONSTR c i r = h4/ind__type/BOTTOM)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Goal: !f e. h4/lbtree/lbtree__case e f h4/lbtree/Lf = e
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_Q1287539,TV_Q1287535]: ![V_f, V_g]: (![V_x]: s(TV_Q1287535,happ(s(t_fun(TV_Q1287539,TV_Q1287535),V_f),s(TV_Q1287539,V_x))) = s(TV_Q1287535,happ(s(t_fun(TV_Q1287539,TV_Q1287535),V_g),s(TV_Q1287539,V_x))) => s(t_fun(TV_Q1287539,TV_Q1287535),V_f) = s(t_fun(TV_Q1287539,TV_Q1287535),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_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_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_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_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_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_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_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_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_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_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_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_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_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_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_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_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_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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_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_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_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_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_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_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_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_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_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_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_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_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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
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_DISJu_u_IMPu_u_THM, axiom, ![V_R, V_Q, V_P]: (((p(s(t_bool,V_P)) | p(s(t_bool,V_Q))) => p(s(t_bool,V_R))) <=> ((p(s(t_bool,V_P)) => p(s(t_bool,V_R))) & (p(s(t_bool,V_Q)) => p(s(t_bool,V_R)))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_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_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_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_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_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_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_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_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_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_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_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_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_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_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_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_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_bools_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_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_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_options_optionu_u_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_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_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_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,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_relations_RESTRICTu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_R, V_xi_]: s(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_xi_))) = s(TV_u_27b,h4s_bools_cond(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_xi_))),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_))),s(TV_u_27b,h4s_bools_arb)))).
fof(ah4s_pairs_UNCURRYu_u_CONG, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_M) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_Mu_27) & ![V_x, V_y]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_Mu_27) = 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))) = 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_fu_27),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) => s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_M))) = s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_fu_27))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_Mu_27))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_relations_RSUBSET0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R2, V_R1]: (p(s(t_bool,h4s_relations_rsubset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_DISJu_u_COMM, 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_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_setu_u_relations_relu_u_tou_u_relnu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_R, V_x, V_y]: s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))),s(t_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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) => ![V_uu_0]: (![V_R, V_x]: s(t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27a,V_x))) => ![V_R]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_setu_u_relations_relu_u_tou_u_reln(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R)))))))))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_pairs_produ_u_TYu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_p]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_p)))) <=> ?[V_x, V_y]: ![V_xi_, V_x0]: (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_xi_))),s(TV_u_27b,V_x0)))) <=> (s(TV_u_27a,V_xi_) = s(TV_u_27a,V_x) & s(TV_u_27b,V_x0) = s(TV_u_27b,V_y)))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_rep)))))).
fof(ah4s_pairs_ABSu_u_REPu_u_produ_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_r]: (?[V_x, V_y]: ![V_xi_, V_x0]: (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_r),s(TV_u_27a,V_xi_))),s(TV_u_27b,V_x0)))) <=> (s(TV_u_27a,V_xi_) = s(TV_u_27a,V_x) & s(TV_u_27b,V_x0) = s(TV_u_27b,V_y))) <=> s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_pairs_repu_u_prod(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_absu_u_prod(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_r))).
fof(ah4s_bools_itselfu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_e]: ?[V_f]: s(TV_u_27a,happ(s(t_fun(t_h4s_bools_itself(TV_u_27b),TV_u_27a),V_f),s(t_h4s_bools_itself(TV_u_27b),h4s_bools_theu_u_value))) = s(TV_u_27a,V_e)).
fof(ah4s_relations_TFLu_u_INDUCTIVEu_u_INVARIANTu_u_ONu_u_WFREC, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f, V_R, V_P, V_M, V_D]: ((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)))) & (p(s(t_bool,h4s_relations_inductiveu_u_invariantu_u_on(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_D),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),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,happ(s(t_fun(TV_u_27a,t_bool),V_D),s(TV_u_27a,V_x))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_relations_TCu_u_liftsu_u_equalities, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R]: (![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_x))),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_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))) => ![V_x, V_y]: (p(s(t_bool,h4s_relations_tc(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_y)))) => 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_y)))))).
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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_relations_INDUCTIVEu_u_INVARIANTu_u_ONu_u_WFREC, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_R, V_P, V_M, V_D]: ((p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & (p(s(t_bool,h4s_relations_inductiveu_u_invariantu_u_on(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_D),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),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,happ(s(t_fun(TV_u_27a,t_bool),V_D),s(TV_u_27a,V_x)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),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))),s(TV_u_27a,V_x)))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(ah4s_sums_OUTR0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(TV_u_27b,h4s_sums_outr(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_x))))) = s(TV_u_27b,V_x)).
fof(ah4s_relations_TCu_u_INDUCT, axiom, ![TV_u_27a]: ![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_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, V_z]: ((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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => 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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_u, V_v]: (p(s(t_bool,h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_u),s(TV_u_27a,V_v)))) => 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_P),s(TV_u_27a,V_u))),s(TV_u_27a,V_v))))))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_sums_sumu_u_axiom, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_f, V_g, V_h]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g))),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h)))) <=> (s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl))) = s(t_fun(TV_u_27a,TV_u_27c),V_f) & s(t_fun(TV_u_27b,TV_u_27c),h4s_combins_o(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr))) = s(t_fun(TV_u_27b,TV_u_27c),V_g))) => ![V_g, V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g)))))))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_llists_llistu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: ?[V_g]: (![V_x]: s(t_h4s_options_option(TV_u_27b),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),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))))) & ![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),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(TV_u_27a,V_x))))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b)),h4s_combins_o(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst))),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))))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_llists_llistu_u_ueu_u_Axiom, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f))),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g)))) <=> (![V_x]: s(t_h4s_options_option(TV_u_27b),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),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))))) & ![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),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(TV_u_27a,V_x))))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b)),h4s_combins_o(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst))),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))))))) => ![V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f)))))))).
fof(ah4s_options_optionu_u_CLAUSESu_c14, 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_relations_RDOMu_u_DELETEu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_u, V_R]: (p(s(t_bool,h4s_relations_rdomu_u_delete(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_u),s(TV_u_27b,V_v)))) <=> (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_R),s(TV_u_27a,V_u))),s(TV_u_27b,V_v)))) & ~ (s(TV_u_27a,V_u) = s(TV_u_27a,V_x))))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
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_indu_u_types_CONSTRu_u_IND, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_P),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom)))) & ![V_c, V_i, V_r]: (![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_P),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_P),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r)))))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_P),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_x)))))).
fof(ah4s_indu_u_types_CONSTRu_u_INJ, axiom, ![TV_u_27a]: ![V_r2, V_r1, V_i2, V_i1, V_c2, V_c1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c1),s(TV_u_27a,V_i1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r1))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c2),s(TV_u_27a,V_i2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r2))) <=> (s(t_h4s_nums_num,V_c1) = s(t_h4s_nums_num,V_c2) & (s(TV_u_27a,V_i1) = s(TV_u_27a,V_i2) & s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r1) = s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r2))))).
fof(ah4s_indu_u_types_CONSTRu_u_BOT, axiom, ![TV_u_27a]: ![V_r, V_i, V_c]: ~ (s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ch4s_lbtrees_lbtreeu_u_caseu_u_thmu_c0, conjecture, ![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)).
