%   ORIGINAL: h4/lbtree/exists__bf__flatten
% 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/bf__flatten__def_c1: !ts. h4/lbtree/bf__flatten (h4/list/CONS h4/lbtree/Lf ts) = h4/lbtree/bf__flatten ts
% Assm: h4/bool/TRUTH: T
% Assm: h4/lbtree/bf__flatten__eq__lnil: !l. h4/lbtree/bf__flatten l = h4/llist/LNIL <=> h4/list/EVERY ($equals h4/lbtree/Lf) l
% Assm: h4/lbtree/bf__flatten__append: !l2 l1. h4/list/EVERY ($equals h4/lbtree/Lf) l1 ==> h4/lbtree/bf__flatten (h4/list/APPEND l1 l2) = h4/lbtree/bf__flatten l2
% Assm: h4/list/ALL__DISTINCT__FILTER: !l. h4/list/ALL__DISTINCT l <=> (!x. h4/bool/IN x (h4/list/LIST__TO__SET l) ==> h4/list/FILTER ($equals x) l = h4/list/CONS x h4/list/NIL)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/lbtree/bf__flatten__def_c2: !ts t2 t1 a. h4/lbtree/bf__flatten (h4/list/CONS (h4/lbtree/Nd a t1 t2) ts) = h4/llist/LCONS a (h4/lbtree/bf__flatten (h4/list/APPEND ts (h4/list/CONS t1 (h4/list/CONS t2 h4/list/NIL))))
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/lbtree/mem__def: h4/lbtree/mem = (\a0 a1. !mem_27. (!a00 a10. (?t1 t2. a10 = h4/lbtree/Nd a00 t1 t2) \/ (?b t1 t2. a10 = h4/lbtree/Nd b t1 t2 /\ mem_27 a00 t1) \/ (?b t1 t2. a10 = h4/lbtree/Nd b t1 t2 /\ mem_27 a00 t2) ==> mem_27 a00 a10) ==> mem_27 a0 a1)
% 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/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/lbtree/bf__flatten__def_c0: h4/lbtree/bf__flatten h4/list/NIL = h4/llist/LNIL
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/lbtree/Lf__NOT__Nd: !t2 t1 a. ~(h4/lbtree/Lf = h4/lbtree/Nd a t1 t2)
% 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/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/list/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/lbtree/drop__while__def_c1: !t h P. h4/lbtree/drop__while P (h4/list/CONS h t) = h4/bool/COND (P h) (h4/lbtree/drop__while P t) (h4/list/CONS h t)
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/llist/llist__Axiom__1: !f. ?g. !x. g x = h4/option/option__CASE (f x) h4/llist/LNIL (\v. h4/pair/pair__CASE v (\a b. h4/llist/LCONS b (g a)))
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/lbtree/lbtree__case__thm_c1: !t2 t1 f e a. h4/lbtree/lbtree__case e f (h4/lbtree/Nd a t1 t2) = f a t1 t2
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/llist/LCONS__11: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm: h4/lbtree/drop__while__def_c0: !P. h4/lbtree/drop__while P h4/list/NIL = h4/list/NIL
% 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/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/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/llist/exists__def: h4/llist/exists = (\P a0. !exists_27. (!a00. (?h t. a00 = h4/llist/LCONS h t /\ P h) \/ (?h t. a00 = h4/llist/LCONS h t /\ exists_27 t) ==> exists_27 a00) ==> exists_27 a0)
% Assm: h4/lbtree/mem__thm_c0: !a. h4/lbtree/mem a h4/lbtree/Lf <=> F
% Assm: h4/lbtree/mem__rules_c0: !t2 t1 a. h4/lbtree/mem a (h4/lbtree/Nd a t1 t2)
% Assm: h4/lbtree/mem__ind: !mem_27. (!a t1 t2. mem_27 a (h4/lbtree/Nd a t1 t2)) /\ (!a b t1 t2. mem_27 a t1 ==> mem_27 a (h4/lbtree/Nd b t1 t2)) /\ (!a b t1 t2. mem_27 a t2 ==> mem_27 a (h4/lbtree/Nd b t1 t2)) ==> (!a0 a1. h4/lbtree/mem a0 a1 ==> mem_27 a0 a1)
% Assm: h4/lbtree/mem__strongind: !mem_27. (!a t1 t2. mem_27 a (h4/lbtree/Nd a t1 t2)) /\ (!a b t1 t2. h4/lbtree/mem a t1 /\ mem_27 a t1 ==> mem_27 a (h4/lbtree/Nd b t1 t2)) /\ (!a b t1 t2. h4/lbtree/mem a t2 /\ mem_27 a t2 ==> mem_27 a (h4/lbtree/Nd b t1 t2)) ==> (!a0 a1. h4/lbtree/mem a0 a1 ==> mem_27 a0 a1)
% Assm: h4/lbtree/mem__cases: !a1 a0. h4/lbtree/mem a0 a1 <=> (?t1 t2. a1 = h4/lbtree/Nd a0 t1 t2) \/ (?b t1 t2. a1 = h4/lbtree/Nd b t1 t2 /\ h4/lbtree/mem a0 t1) \/ (?b t1 t2. a1 = h4/lbtree/Nd b t1 t2 /\ h4/lbtree/mem a0 t2)
% Assm: h4/lbtree/mem__thm_c1: !t2 t1 b a. h4/lbtree/mem a (h4/lbtree/Nd b t1 t2) <=> a = b \/ h4/lbtree/mem a t1 \/ h4/lbtree/mem a t2
% Assm: h4/lbtree/mem__rules_c2: !t2 t1 b a. h4/lbtree/mem a t2 ==> h4/lbtree/mem a (h4/lbtree/Nd b t1 t2)
% Assm: h4/lbtree/mem__rules_c1: !t2 t1 b a. h4/lbtree/mem a t1 ==> h4/lbtree/mem a (h4/lbtree/Nd b t1 t2)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/combin/literal__case__FORALL__ELIM: !v f. h4/bool/literal__case f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm: h4/combin/LET__FORALL__ELIM: !v f. h4/bool/LET f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm: h4/relation/O__Id: !R. h4/relation/O R $equals = R
% Assm: h4/relation/inv__Id: h4/relation/inv $equals = $equals
% Assm: h4/relation/Id__O: !R. h4/relation/O $equals R = R
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/lbtree/lbtree__cases: !t. t = h4/lbtree/Lf \/ (?a t1 t2. t = h4/lbtree/Nd a t1 t2)
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/llist/LCONS__NOT__NIL_c0: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/relation/O__DEF: !z x R2 R1. h4/relation/O R1 R2 x z <=> (?y. R2 x y /\ R1 y z)
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/list/ALL__DISTINCT0_c1: !t h. h4/list/ALL__DISTINCT (h4/list/CONS h t) <=> ~h4/bool/IN h (h4/list/LIST__TO__SET t) /\ h4/list/ALL__DISTINCT t
% Assm: h4/list/ALL__DISTINCT0_c0: h4/list/ALL__DISTINCT h4/list/NIL <=> T
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/list/MEM_c0: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm: h4/list/NOT__CONS__NIL: !a1 a0. ~(h4/list/CONS a0 a1 = h4/list/NIL)
% Assm: h4/list/MEM_c1: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/FILTER0_c1: !t h P. h4/list/FILTER P (h4/list/CONS h t) = h4/bool/COND (P h) (h4/list/CONS h (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm: h4/list/FILTER0_c0: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/LEFT__AND__OVER__OR: !C B A. A /\ (B \/ C) <=> A /\ B \/ A /\ C
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/COND__EXPAND: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/COND__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Assm: h4/bool/DISJ__IMP__THM: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm: h4/bool/COND__RATOR: !x g f b. h4/bool/COND b f g x = h4/bool/COND b (f x) (g x)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/relation/inv__DEF: !y x R. h4/relation/inv R x y <=> R y x
% Assm: h4/bool/literal__case__THM: !x f. h4/bool/literal__case f x = f x
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% 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
% Goal: !x tlist. h4/llist/exists ($equals x) (h4/lbtree/bf__flatten tlist) ==> h4/list/EXISTS (h4/lbtree/mem x) tlist
%   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_bfu_u_flattenu_u_defu_c1]: !ts. h4/lbtree/bf__flatten (h4/list/CONS h4/lbtree/Lf ts) = h4/lbtree/bf__flatten ts
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_lbtrees_bfu_u_flattenu_u_equ_u_lnil]: !l. h4/lbtree/bf__flatten l = h4/llist/LNIL <=> h4/list/EVERY (happ $equals h4/lbtree/Lf) l
% Assm [h4s_lbtrees_bfu_u_flattenu_u_append]: !l2 l1. h4/list/EVERY (happ $equals h4/lbtree/Lf) l1 ==> h4/lbtree/bf__flatten (h4/list/APPEND l1 l2) = h4/lbtree/bf__flatten l2
% Assm [h4s_lists_ALLu_u_DISTINCTu_u_FILTER]: !l. h4/list/ALL__DISTINCT l <=> (!x. h4/bool/IN x (h4/list/LIST__TO__SET l) ==> h4/list/FILTER (happ $equals x) l = h4/list/CONS x h4/list/NIL)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_lbtrees_bfu_u_flattenu_u_defu_c2]: !ts t2 t1 a. h4/lbtree/bf__flatten (h4/list/CONS (h4/lbtree/Nd a t1 t2) ts) = h4/llist/LCONS a (h4/lbtree/bf__flatten (h4/list/APPEND ts (h4/list/CONS t1 (h4/list/CONS t2 h4/list/NIL))))
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_lbtrees_memu_u_def]: !x x. happ (h4/lbtree/mem x) x <=> (!mem_27. (!a00 a10. (?t1 t2. a10 = h4/lbtree/Nd a00 t1 t2) \/ (?b t1 t2. a10 = h4/lbtree/Nd b t1 t2 /\ happ (happ mem_27 a00) t1) \/ (?b t1 t2. a10 = h4/lbtree/Nd b t1 t2 /\ happ (happ mem_27 a00) t2) ==> happ (happ mem_27 a00) a10) ==> happ (happ mem_27 x) x)
% 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_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_lbtrees_bfu_u_flattenu_u_defu_c0]: h4/lbtree/bf__flatten h4/list/NIL = h4/llist/LNIL
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_lbtrees_Lfu_u_NOTu_u_Nd]: !t2 t1 a. ~(h4/lbtree/Lf = h4/lbtree/Nd a t1 t2)
% 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_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_lists_EVERYu_u_DEFu_c1]: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> happ P h /\ h4/list/EVERY P t
% Assm [h4s_lists_EVERYu_u_DEFu_c0]: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_lbtrees_dropu_u_whileu_u_defu_c1]: !t h P. h4/lbtree/drop__while P (h4/list/CONS h t) = h4/bool/COND (happ P h) (h4/lbtree/drop__while P t) (h4/list/CONS h t)
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_llists_llistu_u_Axiomu_u_1]: !_2. (!g a b. happ (happ (happ _2 g) a) b = h4/llist/LCONS b (happ g a)) ==> (!_1. (!g a. happ (happ _1 g) a = happ (happ _2 g) a) ==> (!_0. (!g v. happ (happ _0 g) v = h4/pair/pair__CASE v (happ _1 g)) ==> (!f. ?g. !x. happ g x = h4/option/option__CASE (happ f x) h4/llist/LNIL (happ _0 g))))
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_lbtrees_lbtreeu_u_caseu_u_thmu_c1]: !t2 t1 f e a. h4/lbtree/lbtree__case e f (h4/lbtree/Nd a t1 t2) = happ (happ (happ f a) t1) t2
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_llists_LCONSu_u_11]: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm [h4s_lbtrees_dropu_u_whileu_u_defu_c0]: !P. h4/lbtree/drop__while P h4/list/NIL = h4/list/NIL
% 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_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_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_llists_existsu_u_def]: !x x. h4/llist/exists x x <=> (!exists_27. (!a00. (?h t. a00 = h4/llist/LCONS h t /\ happ x h) \/ (?h t. a00 = h4/llist/LCONS h t /\ happ exists_27 t) ==> happ exists_27 a00) ==> happ exists_27 x)
% Assm [h4s_lbtrees_memu_u_thmu_c0]: !a. happ (h4/lbtree/mem a) h4/lbtree/Lf <=> F
% Assm [h4s_lbtrees_memu_u_rulesu_c0]: !t2 t1 a. happ (h4/lbtree/mem a) (h4/lbtree/Nd a t1 t2)
% Assm [h4s_lbtrees_memu_u_ind]: !mem_27. (!a t1 t2. happ (happ mem_27 a) (h4/lbtree/Nd a t1 t2)) /\ (!a b t1 t2. happ (happ mem_27 a) t1 ==> happ (happ mem_27 a) (h4/lbtree/Nd b t1 t2)) /\ (!a b t1 t2. happ (happ mem_27 a) t2 ==> happ (happ mem_27 a) (h4/lbtree/Nd b t1 t2)) ==> (!a0 a1. happ (h4/lbtree/mem a0) a1 ==> happ (happ mem_27 a0) a1)
% Assm [h4s_lbtrees_memu_u_strongind]: !mem_27. (!a t1 t2. happ (happ mem_27 a) (h4/lbtree/Nd a t1 t2)) /\ (!a b t1 t2. happ (h4/lbtree/mem a) t1 /\ happ (happ mem_27 a) t1 ==> happ (happ mem_27 a) (h4/lbtree/Nd b t1 t2)) /\ (!a b t1 t2. happ (h4/lbtree/mem a) t2 /\ happ (happ mem_27 a) t2 ==> happ (happ mem_27 a) (h4/lbtree/Nd b t1 t2)) ==> (!a0 a1. happ (h4/lbtree/mem a0) a1 ==> happ (happ mem_27 a0) a1)
% Assm [h4s_lbtrees_memu_u_cases]: !a1 a0. happ (h4/lbtree/mem a0) a1 <=> (?t1 t2. a1 = h4/lbtree/Nd a0 t1 t2) \/ (?b t1 t2. a1 = h4/lbtree/Nd b t1 t2 /\ happ (h4/lbtree/mem a0) t1) \/ (?b t1 t2. a1 = h4/lbtree/Nd b t1 t2 /\ happ (h4/lbtree/mem a0) t2)
% Assm [h4s_lbtrees_memu_u_thmu_c1]: !t2 t1 b a. happ (h4/lbtree/mem a) (h4/lbtree/Nd b t1 t2) <=> a = b \/ happ (h4/lbtree/mem a) t1 \/ happ (h4/lbtree/mem a) t2
% Assm [h4s_lbtrees_memu_u_rulesu_c2]: !t2 t1 b a. happ (h4/lbtree/mem a) t2 ==> happ (h4/lbtree/mem a) (h4/lbtree/Nd b t1 t2)
% Assm [h4s_lbtrees_memu_u_rulesu_c1]: !t2 t1 b a. happ (h4/lbtree/mem a) t1 ==> happ (h4/lbtree/mem a) (h4/lbtree/Nd b t1 t2)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_combins_literalu_u_caseu_u_FORALLu_u_ELIM]: !v f. h4/bool/literal__case f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm [h4s_combins_LETu_u_FORALLu_u_ELIM]: !v f. h4/bool/LET f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm [h4s_relations_Ou_u_Id]: !R. h4/relation/O R $equals = R
% Assm [h4s_relations_invu_u_Id]: h4/relation/inv $equals = $equals
% Assm [h4s_relations_Idu_u_O]: !R. h4/relation/O $equals R = R
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_lbtrees_lbtreeu_u_cases]: !t. t = h4/lbtree/Lf \/ (?a t1 t2. t = h4/lbtree/Nd a t1 t2)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_llists_LCONSu_u_NOTu_u_NILu_c0]: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_relations_Ou_u_DEF]: !z x R2 R1. happ (happ (h4/relation/O R1 R2) x) z <=> (?y. happ (happ R2 x) y /\ happ (happ R1 y) z)
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Su_u_DEF]: !x x x. happ (h4/combin/S x x) x = happ (happ x x) (happ x x)
% Assm [h4s_combins_Cu_u_DEF]: !x x x. happ (h4/combin/C x x) x = happ (happ x x) x
% Assm [h4s_markers_Abbrevu_u_def]: !x. happ h4/marker/Abbrev x <=> x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_lists_ALLu_u_DISTINCT0u_c1]: !t h. h4/list/ALL__DISTINCT (h4/list/CONS h t) <=> ~h4/bool/IN h (h4/list/LIST__TO__SET t) /\ h4/list/ALL__DISTINCT t
% Assm [h4s_lists_ALLu_u_DISTINCT0u_c0]: h4/list/ALL__DISTINCT h4/list/NIL <=> T
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lists_MEMu_c0]: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm [h4s_lists_NOTu_u_CONSu_u_NIL]: !a1 a0. ~(h4/list/CONS a0 a1 = h4/list/NIL)
% Assm [h4s_lists_MEMu_c1]: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_lists_FILTER0u_c1]: !t h P. h4/list/FILTER P (h4/list/CONS h t) = h4/bool/COND (happ P h) (h4/list/CONS h (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm [h4s_lists_FILTER0u_c0]: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% 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_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_LEFTu_u_ANDu_u_OVERu_u_OR]: !C B A. A /\ (B \/ C) <=> A /\ B \/ A /\ C
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ 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_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_CONDu_u_EXPAND]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_CONDu_u_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% Assm [h4s_bools_DISJu_u_IMPu_u_THM]: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm [h4s_bools_CONDu_u_RATOR]: !x g f b. happ (h4/bool/COND b f g) x = h4/bool/COND b (happ f x) (happ g x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_relations_invu_u_DEF]: !y x R. happ (happ (h4/relation/inv R) x) y <=> happ (happ R y) x
% Assm [h4s_bools_literalu_u_caseu_u_THM]: !x f. h4/bool/literal__case f x = happ f x
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% 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
% Goal: !x tlist. h4/llist/exists (happ $equals x) (h4/lbtree/bf__flatten tlist) ==> h4/list/EXISTS (h4/lbtree/mem x) tlist
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1288224,TV_Q1288220]: ![V_f, V_g]: (![V_x]: s(TV_Q1288220,happ(s(t_fun(TV_Q1288224,TV_Q1288220),V_f),s(TV_Q1288224,V_x))) = s(TV_Q1288220,happ(s(t_fun(TV_Q1288224,TV_Q1288220),V_g),s(TV_Q1288224,V_x))) => s(t_fun(TV_Q1288224,TV_Q1288220),V_f) = s(t_fun(TV_Q1288224,TV_Q1288220),V_g))).
fof(ah4s_lbtrees_bfu_u_flattenu_u_defu_c1, axiom, ![TV_u_27a]: ![V_ts]: s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),h4s_lists_cons(s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf),s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_ts))))) = s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_ts)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_lbtrees_bfu_u_flattenu_u_equ_u_lnil, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_l))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil) <=> p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),d_equals),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf))),s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_l)))))).
fof(ah4s_lbtrees_bfu_u_flattenu_u_append, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),d_equals),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf))),s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_l1)))) => s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),h4s_lists_append(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_l1),s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_l2))))) = s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_l2))))).
fof(ah4s_lists_ALLu_u_DISTINCTu_u_FILTER, axiom, ![TV_u_27a]: ![V_l]: (p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_l)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(TV_u_27a,V_x))),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_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))))).
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_lbtrees_bfu_u_flattenu_u_defu_c2, axiom, ![TV_u_27a]: ![V_ts, V_t2, V_t1, V_a]: s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),h4s_lists_cons(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_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_ts))))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_a),s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),h4s_lists_append(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_ts),s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),h4s_lists_cons(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),h4s_lists_cons(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2),s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),h4s_lists_nil)))))))))))).
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_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_lbtrees_memu_u_def, axiom, ![TV_u_27a]: ![V_x, V_x0]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_x0)))) <=> ![V_memu_27]: (![V_a00, V_a10]: ((?[V_t1, V_t2]: s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a10) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a00),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) | (?[V_b, V_t1, V_t2]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a10) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) & p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a00))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))))) | ?[V_b, V_t1, V_t2]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a10) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) & p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a00))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a00))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a10))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_x0))))))).
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_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_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_lbtrees_bfu_u_flattenu_u_defu_c0, axiom, ![TV_u_27a]: s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),h4s_lists_nil))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil)).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_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_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_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_lists_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(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)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_EVERYu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_IMPu_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_lbtrees_dropu_u_whileu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lbtrees_dropu_u_while(s(t_fun(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))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lbtrees_dropu_u_while(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))),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_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_llists_llistu_u_Axiomu_u_1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_g, V_a, V_b]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))),s(TV_u_27a,V_a))),s(TV_u_27b,V_b))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_b),s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(TV_u_27a,V_a))))) => ![V_uu_1]: (![V_g, V_a]: s(t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))),s(TV_u_27a,V_a))) = s(t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))),s(TV_u_27a,V_a))) => ![V_uu_0]: (![V_g, V_v]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))) = s(t_h4s_llists_llist(TV_u_27b),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_llists_llist(TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))))) => ![V_f]: ?[V_g]: ![V_x]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g),s(TV_u_27a,V_x))) = s(t_h4s_llists_llist(TV_u_27b),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27a,V_x))),s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_llists_llist(TV_u_27b))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_llists_llist(TV_u_27b)),V_g))))))))).
fof(ah4s_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_lbtrees_lbtreeu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_t2, V_t1, V_f, V_e, V_a]: s(TV_u_27a,h4s_lbtrees_lbtreeu_u_case(s(TV_u_27a,V_e),s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(TV_u_27b,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t1))),s(t_h4s_lbtrees_lbtree(TV_u_27b),V_t2)))).
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_llists_LCONSu_u_11, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_h2, V_h1]: (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h1),s(t_h4s_llists_llist(TV_u_27a),V_t1))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h2),s(t_h4s_llists_llist(TV_u_27a),V_t2))) <=> (s(TV_u_27a,V_h1) = s(TV_u_27a,V_h2) & s(t_h4s_llists_llist(TV_u_27a),V_t1) = s(t_h4s_llists_llist(TV_u_27a),V_t2)))).
fof(ah4s_lbtrees_dropu_u_whileu_u_defu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lbtrees_dropu_u_while(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_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_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_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
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_llists_existsu_u_def, axiom, ![TV_u_27a]: ![V_x, V_x0]: (p(s(t_bool,h4s_llists_exists(s(t_fun(TV_u_27a,t_bool),V_x),s(t_h4s_llists_llist(TV_u_27a),V_x0)))) <=> ![V_existsu_27]: (![V_a00]: ((?[V_h, V_t]: (s(t_h4s_llists_llist(TV_u_27a),V_a00) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_h))))) | ?[V_h, V_t]: (s(t_h4s_llists_llist(TV_u_27a),V_a00) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))) & p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_existsu_27),s(t_h4s_llists_llist(TV_u_27a),V_t)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_existsu_27),s(t_h4s_llists_llist(TV_u_27a),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_bool),V_existsu_27),s(t_h4s_llists_llist(TV_u_27a),V_x0))))))).
fof(ah4s_lbtrees_memu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf))) = s(t_bool,f)).
fof(ah4s_lbtrees_memu_u_rulesu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_a]: p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,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))))))).
fof(ah4s_lbtrees_memu_u_ind, axiom, ![TV_u_27a]: ![V_memu_27]: ((![V_a, V_t1, V_t2]: p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,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)))))) & (![V_a, V_b, V_t1, V_t2]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))) & ![V_a, V_b, V_t1, V_t2]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))))) => ![V_a0, V_a1]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1))))))).
fof(ah4s_lbtrees_memu_u_strongind, axiom, ![TV_u_27a]: ![V_memu_27]: ((![V_a, V_t1, V_t2]: p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,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)))))) & (![V_a, V_b, V_t1, V_t2]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1)))) & p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))) & ![V_a, V_b, V_t1, V_t2]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))))) => ![V_a0, V_a1]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool)),V_memu_27),s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1))))))).
fof(ah4s_lbtrees_memu_u_cases, axiom, ![TV_u_27a]: ![V_a1, V_a0]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1)))) <=> (?[V_t1, V_t2]: s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a0),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) | (?[V_b, V_t1, V_t2]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) & p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))))) | ?[V_b, V_t1, V_t2]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) & p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))))).
fof(ah4s_lbtrees_memu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))))) <=> (s(TV_u_27a,V_a) = s(TV_u_27a,V_b) | (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1)))) | p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))))))).
fof(ah4s_lbtrees_memu_u_rulesu_c2, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))))))).
fof(ah4s_lbtrees_memu_u_rulesu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_b, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_a))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_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_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_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_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_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_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_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_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_combins_literalu_u_caseu_u_FORALLu_u_ELIM, axiom, ![TV_u_27a]: ![V_v, V_f]: s(t_bool,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v))) = s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),h4s_combins_o(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_imply),s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),h4s_markers_abbrev),s(t_fun(TV_u_27a,t_bool),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(TV_u_27a,V_v))))))),s(t_fun(TV_u_27a,t_bool),V_f)))))).
fof(ah4s_combins_LETu_u_FORALLu_u_ELIM, axiom, ![TV_u_27a]: ![V_v, V_f]: s(t_bool,h4s_bools_let(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v))) = s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),h4s_combins_o(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_imply),s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),h4s_markers_abbrev),s(t_fun(TV_u_27a,t_bool),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(TV_u_27a,V_v))))))),s(t_fun(TV_u_27a,t_bool),V_f)))))).
fof(ah4s_relations_Ou_u_Id, axiom, ![TV_u_27a,TV_u_27b]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_o(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_27a,t_bool)),d_equals))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R)).
fof(ah4s_relations_invu_u_Id, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals)).
fof(ah4s_relations_Idu_u_O, axiom, ![TV_u_27a,TV_u_27b]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_o(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),d_equals),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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_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_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_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_llists_LCONSu_u_NOTu_u_NILu_c0, axiom, ![TV_u_27a]: ![V_t, V_h]: ~ (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
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_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_relations_Ou_u_DEF, axiom, ![TV_u_27g,TV_u_27h,TV_u_27k]: ![V_z, V_x, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(TV_u_27k,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27k,t_bool)),h4s_relations_o(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2))),s(TV_u_27g,V_x))),s(TV_u_27k,V_z)))) <=> ?[V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27h,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2),s(TV_u_27g,V_x))),s(TV_u_27h,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27k,t_bool),happ(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(TV_u_27h,V_y))),s(TV_u_27k,V_z))))))).
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_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0))),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27b,V_x0))),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),h4s_markers_abbrev),s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_lists_ALLu_u_DISTINCT0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: (p(s(t_bool,h4s_lists_allu_u_distinct(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)))))) <=> (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_h),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_t))))))) & p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_ALLu_u_DISTINCT0u_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
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,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lists_MEMu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))) = s(t_bool,f)).
fof(ah4s_lists_NOTu_u_CONSu_u_NIL, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (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(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))).
fof(ah4s_lists_MEMu_c1, axiom, ![TV_u_27a]: ![V_x, V_t, V_h]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(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)))))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_h) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_t))))))))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (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(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0u_27),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_lists_listu_u_INDUCT, 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_lists_FILTER0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(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))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))),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),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_FILTER0u_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
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_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_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_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_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_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_LEFTu_u_ANDu_u_OVERu_u_OR, 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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_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_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_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_EXPAND, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((~ (p(s(t_bool,V_b))) | p(s(t_bool,V_t1))) & (p(s(t_bool,V_b)) | p(s(t_bool,V_t2)))))).
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_bools_CONDu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),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_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_CONDu_u_RATOR, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),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_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_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_relations_invu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: 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)),h4s_relations_inv(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(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))).
fof(ah4s_bools_literalu_u_caseu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_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(ch4s_lbtrees_existsu_u_bfu_u_flatten, conjecture, ![TV_u_27a]: ![V_x, V_tlist]: (p(s(t_bool,h4s_llists_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(TV_u_27a,V_x))),s(t_h4s_llists_llist(TV_u_27a),h4s_lbtrees_bfu_u_flatten(s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_tlist)))))) => p(s(t_bool,h4s_lists_exists(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_bool),h4s_lbtrees_mem(s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_h4s_lbtrees_lbtree(TV_u_27a)),V_tlist)))))).
