%   ORIGINAL: h4/lbtree/mindepth__thm_c1
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/lbtree/mindepth__def: !x t. h4/lbtree/mindepth x t = h4/bool/COND (h4/lbtree/mem x t) (h4/option/SOME (h4/while/LEAST (\n. h4/lbtree/depth x t n))) h4/option/NONE
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/TRUTH: T
% Assm: h4/lbtree/mindepth__thm_c0: !x. h4/lbtree/mindepth x h4/lbtree/Lf = h4/option/NONE
% Assm: h4/while/OLEAST__EQNS_c3: h4/while/OLEAST (\n. T) = h4/option/SOME h4/num/0
% Assm: h4/llist/LLENGTH__THM_c0: h4/llist/LLENGTH h4/llist/LNIL = h4/option/SOME h4/num/0
% Assm: h4/lbtree/optmin__curried__def: !x1 x. h4/lbtree/optmin x x1 = h4/lbtree/optmin__tupled (h4/pair/_2C x x1)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/lbtree/depth__def: h4/lbtree/depth = (\a0 a1 a2. !depth_27. (!a00 a10 a20. (?t1 t2. a10 = h4/lbtree/Nd a00 t1 t2 /\ a20 = h4/num/0) \/ (?m a t1 t2. a10 = h4/lbtree/Nd a t1 t2 /\ a20 = h4/num/SUC m /\ depth_27 a00 t1 m) \/ (?m a t1 t2. a10 = h4/lbtree/Nd a t1 t2 /\ a20 = h4/num/SUC m /\ depth_27 a00 t2 m) ==> depth_27 a00 a10 a20) ==> depth_27 a0 a1 a2)
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/lbtree/optmin__tupled__primitive__def: h4/lbtree/optmin__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\optmin__tupled a. h4/pair/pair__CASE a (\v v1. h4/option/option__CASE v (h4/option/option__CASE v1 (h4/combin/I h4/option/NONE) (\y. h4/combin/I (h4/option/SOME y))) (\x. h4/option/option__CASE v1 (h4/combin/I (h4/option/SOME x)) (\y_27. h4/combin/I (h4/option/SOME (h4/arithmetic/MIN x y_27))))))
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% 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/lbtree/optmin__def_c3: !y x. h4/lbtree/optmin (h4/option/SOME x) (h4/option/SOME y) = h4/option/SOME (h4/arithmetic/MIN x y)
% Assm: h4/lbtree/optmin__def_c1: !x. h4/lbtree/optmin (h4/option/SOME x) h4/option/NONE = h4/option/SOME x
% Assm: h4/lbtree/optmin__def_c2: !y. h4/lbtree/optmin h4/option/NONE (h4/option/SOME y) = h4/option/SOME y
% Assm: h4/lbtree/optmin__def_c0: h4/lbtree/optmin h4/option/NONE h4/option/NONE = h4/option/NONE
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/lbtree/mem__thm_c0: !a. h4/lbtree/mem a h4/lbtree/Lf <=> F
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% 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/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/lbtree/lbtree__cases: !t. t = h4/lbtree/Lf \/ (?a t1 t2. t = h4/lbtree/Nd a t1 t2)
% Assm: h4/lbtree/lbtree__bisimulation: !u t. t = u <=> (?R. R t u /\ (!t0 u0. R t0 u0 ==> t0 = h4/lbtree/Lf /\ u0 = h4/lbtree/Lf \/ (?a t1 u1 t2 u2. R t1 u1 /\ R t2 u2 /\ t0 = h4/lbtree/Nd a t1 t2 /\ u0 = h4/lbtree/Nd a u1 u2)))
% Assm: h4/lbtree/depth__cases: !a2 a1 a0. h4/lbtree/depth a0 a1 a2 <=> (?t1 t2. a1 = h4/lbtree/Nd a0 t1 t2 /\ a2 = h4/num/0) \/ (?m a t1 t2. a1 = h4/lbtree/Nd a t1 t2 /\ a2 = h4/num/SUC m /\ h4/lbtree/depth a0 t1 m) \/ (?m a t1 t2. a1 = h4/lbtree/Nd a t1 t2 /\ a2 = h4/num/SUC m /\ h4/lbtree/depth a0 t2 m)
% Assm: h4/lbtree/depth__strongind: !depth_27. (!x t1 t2. depth_27 x (h4/lbtree/Nd x t1 t2) h4/num/0) /\ (!m x a t1 t2. h4/lbtree/depth x t1 m /\ depth_27 x t1 m ==> depth_27 x (h4/lbtree/Nd a t1 t2) (h4/num/SUC m)) /\ (!m x a t1 t2. h4/lbtree/depth x t2 m /\ depth_27 x t2 m ==> depth_27 x (h4/lbtree/Nd a t1 t2) (h4/num/SUC m)) ==> (!a0 a1 a2. h4/lbtree/depth a0 a1 a2 ==> depth_27 a0 a1 a2)
% Assm: h4/lbtree/depth__ind: !depth_27. (!x t1 t2. depth_27 x (h4/lbtree/Nd x t1 t2) h4/num/0) /\ (!m x a t1 t2. depth_27 x t1 m ==> depth_27 x (h4/lbtree/Nd a t1 t2) (h4/num/SUC m)) /\ (!m x a t1 t2. depth_27 x t2 m ==> depth_27 x (h4/lbtree/Nd a t1 t2) (h4/num/SUC m)) ==> (!a0 a1 a2. h4/lbtree/depth a0 a1 a2 ==> depth_27 a0 a1 a2)
% Assm: h4/lbtree/depth__rules_c2: !x t2 t1 m a. h4/lbtree/depth x t2 m ==> h4/lbtree/depth x (h4/lbtree/Nd a t1 t2) (h4/num/SUC m)
% Assm: h4/lbtree/depth__rules_c1: !x t2 t1 m a. h4/lbtree/depth x t1 m ==> h4/lbtree/depth x (h4/lbtree/Nd a t1 t2) (h4/num/SUC m)
% Assm: h4/lbtree/depth__rules_c0: !x t2 t1. h4/lbtree/depth x (h4/lbtree/Nd x t1 t2) h4/num/0
% 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/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/lbtree/lbtree__strong__bisimulation: !u t. t = u <=> (?R. R t u /\ (!t0 u0. R t0 u0 ==> t0 = u0 \/ (?a t1 u1 t2 u2. R t1 u1 /\ R t2 u2 /\ t0 = h4/lbtree/Nd a t1 t2 /\ u0 = h4/lbtree/Nd a u1 u2)))
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !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/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sat/dc__disj: !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__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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% 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/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/SELECT__REFL: !x. h4/min/_40 (\y. y = x) = x
% Assm: h4/llist/llength__rel__def: h4/llist/llength__rel = (\a0 a1. !llength__rel_27. (!a00 a10. a00 = h4/llist/LNIL /\ a10 = h4/num/0 \/ (?h n t. a00 = h4/llist/LCONS h t /\ a10 = h4/num/SUC n /\ llength__rel_27 t n) ==> llength__rel_27 a00 a10) ==> llength__rel_27 a0 a1)
% Assm: h4/llist/LCONS__NOT__NIL_c0: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm: h4/llist/LLENGTH0: !ll. h4/llist/LLENGTH ll = h4/bool/COND (h4/llist/LFINITE ll) (h4/option/SOME (h4/min/_40 (\n. h4/llist/llength__rel ll n))) h4/option/NONE
% Assm: h4/llist/LFINITE__THM_c0: h4/llist/LFINITE h4/llist/LNIL <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/while/OLEAST__INTRO: !Q P. ((!n. ~P n) ==> Q h4/option/NONE) /\ (!n. P n /\ (!m. h4/prim__rec/_3C m n ==> ~P m) ==> Q (h4/option/SOME n)) ==> Q (h4/while/OLEAST P)
% Assm: h4/arithmetic/NOT__ZERO__LT__ZERO: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/SOME__DEF: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm: h4/option/option__REP__ABS__DEF_c1: !r. (\x. T) r <=> h4/option/option__REP (h4/option/option__ABS r) = r
% Assm: h4/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> 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/lbtree/Lf__NOT__Nd: !t2 t1 a. ~(h4/lbtree/Lf = h4/lbtree/Nd a t1 t2)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !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/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% 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/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% 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/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% 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/lbtree/drop__while__def_c0: !P. h4/lbtree/drop__while P h4/list/NIL = h4/list/NIL
% 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/llist/LCONS__11: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm: h4/lbtree/lbtree__absrep_c0: !a. h4/lbtree/lbtree__abs (h4/lbtree/lbtree__rep a) = a
% 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/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/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/lbtree/Ndrep__def: !t2 t1 a. h4/lbtree/Ndrep a t1 t2 = (\l. h4/list/list__CASE l (h4/option/SOME a) (\v xs. h4/bool/COND v (t1 xs) (t2 xs)))
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/lbtree/Lf__def: h4/lbtree/Lf = h4/lbtree/lbtree__abs h4/lbtree/Lfrep
% Assm: h4/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/bool/DISJ__IMP__THM: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/lbtree/lbtree__case__thm_c0: !f e. h4/lbtree/lbtree__case e f h4/lbtree/Lf = e
% Assm: h4/bool/SELECT__REFL__2: !x. h4/min/_40 (\y. x = y) = x
% Assm: h4/pair/FORALL__PROD: !P. (!p. P p) <=> (!p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/sat/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm: h4/pair/UNCURRY__ONE__ONE__THM: !g f. h4/pair/UNCURRY f = h4/pair/UNCURRY g <=> f = g
% 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__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Goal: !x t2 t1 a. h4/lbtree/mindepth x (h4/lbtree/Nd a t1 t2) = h4/bool/COND (x = a) (h4/option/SOME h4/num/0) (h4/option/OPTION__MAP h4/num/SUC (h4/lbtree/optmin (h4/lbtree/mindepth x t1) (h4/lbtree/mindepth x t2)))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_lbtrees_mindepthu_u_def]: !_0. (!x t n. happ (happ (happ _0 x) t) n <=> h4/lbtree/depth x t n) ==> (!x t. h4/lbtree/mindepth x t = h4/bool/COND (h4/lbtree/mem x t) (h4/option/SOME (h4/while/LEAST (happ (happ _0 x) t))) h4/option/NONE)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_lbtrees_mindepthu_u_thmu_c0]: !x. h4/lbtree/mindepth x h4/lbtree/Lf = h4/option/NONE
% Assm [h4s_whiles_OLEASTu_u_EQNSu_c3]: !_0. (!n. happ _0 n <=> T) ==> h4/while/OLEAST _0 = h4/option/SOME h4/num/0
% Assm [h4s_llists_LLENGTHu_u_THMu_c0]: h4/llist/LLENGTH h4/llist/LNIL = h4/option/SOME h4/num/0
% Assm [h4s_lbtrees_optminu_u_curriedu_u_def]: !x1 x. h4/lbtree/optmin x x1 = happ h4/lbtree/optmin__tupled (h4/pair/_2C x x1)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_lbtrees_depthu_u_def]: !x x x. h4/lbtree/depth x x x <=> (!depth_27. (!a00 a10 a20. (?t1 t2. a10 = h4/lbtree/Nd a00 t1 t2 /\ a20 = h4/num/0) \/ (?m a t1 t2. a10 = h4/lbtree/Nd a t1 t2 /\ a20 = happ h4/num/SUC m /\ happ (happ (happ depth_27 a00) t1) m) \/ (?m a t1 t2. a10 = h4/lbtree/Nd a t1 t2 /\ a20 = happ h4/num/SUC m /\ happ (happ (happ depth_27 a00) t2) m) ==> happ (happ (happ depth_27 a00) a10) a20) ==> happ (happ (happ depth_27 x) x) x)
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_lbtrees_optminu_u_tupledu_u_primitiveu_u_def]: !_7. (!x y_27. happ (happ _7 x) y_27 = h4/combin/I (h4/option/SOME (h4/arithmetic/MIN x y_27))) ==> (!_6. (!v1 x. happ (happ _6 v1) x = h4/option/option__CASE v1 (h4/combin/I (h4/option/SOME x)) (happ _7 x)) ==> (!_5. (!y. happ _5 y = h4/combin/I (h4/option/SOME y)) ==> (!_4. (!v v1. happ (happ _4 v) v1 = h4/option/option__CASE v (h4/option/option__CASE v1 (h4/combin/I h4/option/NONE) _5) (happ _6 v1)) ==> (!_3. (!v. happ _3 v = happ _4 v) ==> (!_2. (!a. happ _2 a = h4/pair/pair__CASE a _3) ==> (!_1. (!optmin__tupled. happ _1 optmin__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/lbtree/optmin__tupled = h4/relation/WFREC (h4/min/_40 _0) _1)))))))
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% 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_lbtrees_optminu_u_defu_c3]: !y x. h4/lbtree/optmin (h4/option/SOME x) (h4/option/SOME y) = h4/option/SOME (h4/arithmetic/MIN x y)
% Assm [h4s_lbtrees_optminu_u_defu_c1]: !x. h4/lbtree/optmin (h4/option/SOME x) h4/option/NONE = h4/option/SOME x
% Assm [h4s_lbtrees_optminu_u_defu_c2]: !y. h4/lbtree/optmin h4/option/NONE (h4/option/SOME y) = h4/option/SOME y
% Assm [h4s_lbtrees_optminu_u_defu_c0]: h4/lbtree/optmin h4/option/NONE h4/option/NONE = h4/option/NONE
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lbtrees_memu_u_thmu_c0]: !a. h4/lbtree/mem a h4/lbtree/Lf <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_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_lbtrees_lbtreeu_u_cases]: !t. t = h4/lbtree/Lf \/ (?a t1 t2. t = h4/lbtree/Nd a t1 t2)
% Assm [h4s_lbtrees_lbtreeu_u_bisimulation]: !u t. t = u <=> (?R. happ (happ R t) u /\ (!t0 u0. happ (happ R t0) u0 ==> t0 = h4/lbtree/Lf /\ u0 = h4/lbtree/Lf \/ (?a t1 u1 t2 u2. happ (happ R t1) u1 /\ happ (happ R t2) u2 /\ t0 = h4/lbtree/Nd a t1 t2 /\ u0 = h4/lbtree/Nd a u1 u2)))
% Assm [h4s_lbtrees_depthu_u_cases]: !a2 a1 a0. h4/lbtree/depth a0 a1 a2 <=> (?t1 t2. a1 = h4/lbtree/Nd a0 t1 t2 /\ a2 = h4/num/0) \/ (?m a t1 t2. a1 = h4/lbtree/Nd a t1 t2 /\ a2 = happ h4/num/SUC m /\ h4/lbtree/depth a0 t1 m) \/ (?m a t1 t2. a1 = h4/lbtree/Nd a t1 t2 /\ a2 = happ h4/num/SUC m /\ h4/lbtree/depth a0 t2 m)
% Assm [h4s_lbtrees_depthu_u_strongind]: !depth_27. (!x t1 t2. happ (happ (happ depth_27 x) (h4/lbtree/Nd x t1 t2)) h4/num/0) /\ (!m x a t1 t2. h4/lbtree/depth x t1 m /\ happ (happ (happ depth_27 x) t1) m ==> happ (happ (happ depth_27 x) (h4/lbtree/Nd a t1 t2)) (happ h4/num/SUC m)) /\ (!m x a t1 t2. h4/lbtree/depth x t2 m /\ happ (happ (happ depth_27 x) t2) m ==> happ (happ (happ depth_27 x) (h4/lbtree/Nd a t1 t2)) (happ h4/num/SUC m)) ==> (!a0 a1 a2. h4/lbtree/depth a0 a1 a2 ==> happ (happ (happ depth_27 a0) a1) a2)
% Assm [h4s_lbtrees_depthu_u_ind]: !depth_27. (!x t1 t2. happ (happ (happ depth_27 x) (h4/lbtree/Nd x t1 t2)) h4/num/0) /\ (!m x a t1 t2. happ (happ (happ depth_27 x) t1) m ==> happ (happ (happ depth_27 x) (h4/lbtree/Nd a t1 t2)) (happ h4/num/SUC m)) /\ (!m x a t1 t2. happ (happ (happ depth_27 x) t2) m ==> happ (happ (happ depth_27 x) (h4/lbtree/Nd a t1 t2)) (happ h4/num/SUC m)) ==> (!a0 a1 a2. h4/lbtree/depth a0 a1 a2 ==> happ (happ (happ depth_27 a0) a1) a2)
% Assm [h4s_lbtrees_depthu_u_rulesu_c2]: !x t2 t1 m a. h4/lbtree/depth x t2 m ==> h4/lbtree/depth x (h4/lbtree/Nd a t1 t2) (happ h4/num/SUC m)
% Assm [h4s_lbtrees_depthu_u_rulesu_c1]: !x t2 t1 m a. h4/lbtree/depth x t1 m ==> h4/lbtree/depth x (h4/lbtree/Nd a t1 t2) (happ h4/num/SUC m)
% Assm [h4s_lbtrees_depthu_u_rulesu_c0]: !x t2 t1. h4/lbtree/depth x (h4/lbtree/Nd x t1 t2) h4/num/0
% Assm [h4s_lbtrees_memu_u_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 [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_lbtrees_lbtreeu_u_strongu_u_bisimulation]: !u t. t = u <=> (?R. happ (happ R t) u /\ (!t0 u0. happ (happ R t0) u0 ==> t0 = u0 \/ (?a t1 u1 t2 u2. happ (happ R t1) u1 /\ happ (happ R t2) u2 /\ t0 = h4/lbtree/Nd a t1 t2 /\ u0 = h4/lbtree/Nd a u1 u2)))
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% 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_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sats_dcu_u_disj]: !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_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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% 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_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_SELECTu_u_REFL]: !_0. (!x y. happ (happ _0 x) y <=> y = x) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_llists_llengthu_u_relu_u_def]: !x x. h4/llist/llength__rel x x <=> (!llength__rel_27. (!a00 a10. a00 = h4/llist/LNIL /\ a10 = h4/num/0 \/ (?h n t. a00 = h4/llist/LCONS h t /\ a10 = happ h4/num/SUC n /\ happ (happ llength__rel_27 t) n) ==> happ (happ llength__rel_27 a00) a10) ==> happ (happ llength__rel_27 x) x)
% Assm [h4s_llists_LCONSu_u_NOTu_u_NILu_c0]: !t h. ~(h4/llist/LCONS h t = h4/llist/LNIL)
% Assm [h4s_llists_LLENGTH0]: !_0. (!ll n. happ (happ _0 ll) n <=> h4/llist/llength__rel ll n) ==> (!ll. h4/llist/LLENGTH ll = h4/bool/COND (h4/llist/LFINITE ll) (h4/option/SOME (h4/min/_40 (happ _0 ll))) h4/option/NONE)
% Assm [h4s_llists_LFINITEu_u_THMu_c0]: h4/llist/LFINITE h4/llist/LNIL <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_whiles_OLEASTu_u_INTRO]: !Q P. ((!n. ~happ P n) ==> happ Q h4/option/NONE) /\ (!n. happ P n /\ (!m. h4/prim__rec/_3C m n ==> ~happ P m) ==> happ Q (h4/option/SOME n)) ==> happ Q (h4/while/OLEAST P)
% Assm [h4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO]: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_DEF]: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm [h4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c1]: !r. T <=> h4/option/option__REP (h4/option/option__ABS r) = r
% Assm [h4s_sums_INRu_u_INLu_u_11u_c0]: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_lbtrees_memu_u_def]: !x x. 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_lbtrees_Lfu_u_NOTu_u_Nd]: !t2 t1 a. ~(h4/lbtree/Lf = h4/lbtree/Nd a t1 t2)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !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_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 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_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% 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_bools_ANDu_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_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_lbtrees_dropu_u_whileu_u_defu_c0]: !P. h4/lbtree/drop__while P h4/list/NIL = h4/list/NIL
% 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_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_lbtreeu_u_absrepu_c0]: !a. h4/lbtree/lbtree__abs (h4/lbtree/lbtree__rep a) = a
% Assm [h4s_lbtrees_lbtreeu_u_absrepu_c1]: !r. h4/lbtree/is__lbtree r <=> h4/lbtree/lbtree__rep (h4/lbtree/lbtree__abs r) = r
% Assm [h4s_lbtrees_isu_u_lbtreeu_u_def]: !t. 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_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_lbtrees_Ndrepu_u_def]: !_1. (!v t1 t2 xs. happ (happ (happ (happ _1 v) t1) t2) xs = h4/bool/COND v (happ t1 xs) (happ t2 xs)) ==> (!_0. (!t1 t2 v. happ (happ (happ _0 t1) t2) v = happ (happ (happ _1 v) t1) t2) ==> (!t2 t1 a x. happ (h4/lbtree/Ndrep a t1 t2) x = h4/list/list__CASE x (h4/option/SOME a) (happ (happ _0 t1) t2)))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_lbtrees_Lfu_u_def]: h4/lbtree/Lf = h4/lbtree/lbtree__abs h4/lbtree/Lfrep
% Assm [h4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. happ P x ==> Q) <=> (?x. happ P x) ==> Q
% Assm [h4s_bools_DISJu_u_IMPu_u_THM]: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lbtrees_lbtreeu_u_caseu_u_thmu_c0]: !f e. h4/lbtree/lbtree__case e f h4/lbtree/Lf = e
% Assm [h4s_bools_SELECTu_u_REFLu_u_2]: !_0. (!x y. happ (happ _0 x) y <=> x = y) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_pairs_FORALLu_u_PROD]: !P. (!p. happ P p) <=> (!p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% 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_sats_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm [h4s_pairs_UNCURRYu_u_ONEu_u_ONEu_u_THM]: !g f. h4/pair/UNCURRY f = h4/pair/UNCURRY g <=> f = g
% 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_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% Goal: !x t2 t1 a. ?v. (v <=> x = a) /\ h4/lbtree/mindepth x (h4/lbtree/Nd a t1 t2) = h4/bool/COND v (h4/option/SOME h4/num/0) (h4/option/OPTION__MAP h4/num/SUC (h4/lbtree/optmin (h4/lbtree/mindepth x t1) (h4/lbtree/mindepth x t2)))
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_Q1288626,TV_Q1288622]: ![V_f, V_g]: (![V_x]: s(TV_Q1288622,happ(s(t_fun(TV_Q1288626,TV_Q1288622),V_f),s(TV_Q1288626,V_x))) = s(TV_Q1288622,happ(s(t_fun(TV_Q1288626,TV_Q1288622),V_g),s(TV_Q1288626,V_x))) => s(t_fun(TV_Q1288626,TV_Q1288622),V_f) = s(t_fun(TV_Q1288626,TV_Q1288622),V_g))).
fof(ah4s_lbtrees_mindepthu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_t, V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_uu_0),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t),s(t_h4s_nums_num,V_n))) => ![V_x, V_t]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_mindepth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_lbtrees_mem(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_uu_0),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_lbtrees_mindepthu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_mindepth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)).
fof(ah4s_whiles_OLEASTu_u_EQNSu_c3, axiom, ![V_uu_0]: (![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_bool,t) => s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(s(t_fun(t_h4s_nums_num,t_bool),V_uu_0))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_llists_LLENGTHu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_lbtrees_optminu_u_curriedu_u_def, axiom, ![V_x1, V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),V_x),s(t_h4s_options_option(t_h4s_nums_num),V_x1))) = s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),h4s_lbtrees_optminu_u_tupled),s(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_options_option(t_h4s_nums_num),V_x),s(t_h4s_options_option(t_h4s_nums_num),V_x1)))))).
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_depthu_u_def, axiom, ![TV_u_27a]: ![V_x, V_x0, V_x1]: (p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_x0),s(t_h4s_nums_num,V_x1)))) <=> ![V_depthu_27]: (![V_a00, V_a10, V_a20]: ((?[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))) & s(t_h4s_nums_num,V_a20) = s(t_h4s_nums_num,h4s_nums_0)) | (?[V_m, V_a, 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_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) & (s(t_h4s_nums_num,V_a20) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_a00))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))),s(t_h4s_nums_num,V_m)))))) | ?[V_m, V_a, 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_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) & (s(t_h4s_nums_num,V_a20) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_a00))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))),s(t_h4s_nums_num,V_m)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_a00))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a10))),s(t_h4s_nums_num,V_a20))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_x0))),s(t_h4s_nums_num,V_x1))))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_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_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_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_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_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_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_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_lbtrees_optminu_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_7]: (![V_x, V_yu_27]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num))),V_uu_7),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_yu_27))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_combins_i(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_arithmetics_min(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_yu_27))))))) => ![V_uu_6]: (![V_v1, V_x]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num))),V_uu_6),s(t_h4s_options_option(t_h4s_nums_num),V_v1))),s(t_h4s_nums_num,V_x))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_nums_num),V_v1),s(t_h4s_options_option(t_h4s_nums_num),h4s_combins_i(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_x))))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num))),V_uu_7),s(t_h4s_nums_num,V_x))))) => ![V_uu_5]: (![V_y]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),V_uu_5),s(t_h4s_nums_num,V_y))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_combins_i(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_y))))) => ![V_uu_4]: (![V_v, V_v1]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num))),V_uu_4),s(t_h4s_options_option(t_h4s_nums_num),V_v))),s(t_h4s_options_option(t_h4s_nums_num),V_v1))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_nums_num),V_v),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_nums_num),V_v1),s(t_h4s_options_option(t_h4s_nums_num),h4s_combins_i(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),V_uu_5))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num))),V_uu_6),s(t_h4s_options_option(t_h4s_nums_num),V_v1))))) => ![V_uu_3]: (![V_v]: s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num))),V_uu_3),s(t_h4s_options_option(t_h4s_nums_num),V_v))) = s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num))),V_uu_4),s(t_h4s_options_option(t_h4s_nums_num),V_v))) => ![V_uu_2]: (![V_a]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),V_uu_2),s(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),V_a))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),V_a),s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num))),V_uu_3))) => ![V_uu_1]: (![V_optminu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),V_optminu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),h4s_lbtrees_optminu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num))),V_uu_1)))))))))))).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
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_lbtrees_optminu_u_defu_c3, axiom, ![V_y, V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_x))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_y))))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_arithmetics_min(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_y)))))).
fof(ah4s_lbtrees_optminu_u_defu_c1, axiom, ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_x))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_x)))).
fof(ah4s_lbtrees_optminu_u_defu_c2, axiom, ![V_y]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_y))))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_y)))).
fof(ah4s_lbtrees_optminu_u_defu_c0, axiom, s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)).
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_lbtrees_memu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_a]: s(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_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_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_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_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_lbtreeu_u_bisimulation, axiom, ![TV_u_27a]: ![V_u, V_t]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u) <=> ?[V_R]: (p(s(t_bool,happ(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)),V_R),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u)))) & ![V_t0, V_u0]: (p(s(t_bool,happ(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)),V_R),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u0)))) => ((s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t0) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) & s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u0) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf)) | ?[V_a, V_t1, V_u1, V_t2, V_u2]: (p(s(t_bool,happ(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)),V_R),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u1)))) & (p(s(t_bool,happ(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)),V_R),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u2)))) & (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t0) = 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),V_u0) = 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_u1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u2))))))))))).
fof(ah4s_lbtrees_depthu_u_cases, axiom, ![TV_u_27a]: ![V_a2, V_a1, V_a0]: (p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_a0),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1),s(t_h4s_nums_num,V_a2)))) <=> (?[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))) & s(t_h4s_nums_num,V_a2) = s(t_h4s_nums_num,h4s_nums_0)) | (?[V_m, V_a, 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_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) & (s(t_h4s_nums_num,V_a2) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))) & p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_a0),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_nums_num,V_m)))))) | ?[V_m, V_a, 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_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) & (s(t_h4s_nums_num,V_a2) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))) & p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_a0),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2),s(t_h4s_nums_num,V_m)))))))))).
fof(ah4s_lbtrees_depthu_u_strongind, axiom, ![TV_u_27a]: ![V_depthu_27]: ((![V_x, V_t1, V_t2]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))),s(t_h4s_nums_num,h4s_nums_0)))) & (![V_m, V_x, V_a, V_t1, V_t2]: ((p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_nums_num,V_m)))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))),s(t_h4s_nums_num,V_m))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),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_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))))))) & ![V_m, V_x, V_a, V_t1, V_t2]: ((p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2),s(t_h4s_nums_num,V_m)))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))),s(t_h4s_nums_num,V_m))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),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_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))))))))) => ![V_a0, V_a1, V_a2]: (p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_a0),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1),s(t_h4s_nums_num,V_a2)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1))),s(t_h4s_nums_num,V_a2))))))).
fof(ah4s_lbtrees_depthu_u_ind, axiom, ![TV_u_27a]: ![V_depthu_27]: ((![V_x, V_t1, V_t2]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))),s(t_h4s_nums_num,h4s_nums_0)))) & (![V_m, V_x, V_a, V_t1, V_t2]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))),s(t_h4s_nums_num,V_m)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),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_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))))))) & ![V_m, V_x, V_a, V_t1, V_t2]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))),s(t_h4s_nums_num,V_m)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_x))),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_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))))))))) => ![V_a0, V_a1, V_a2]: (p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_a0),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1),s(t_h4s_nums_num,V_a2)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lbtrees_lbtree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool))),V_depthu_27),s(TV_u_27a,V_a0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a1))),s(t_h4s_nums_num,V_a2))))))).
fof(ah4s_lbtrees_depthu_u_rulesu_c2, axiom, ![TV_u_27a]: ![V_x, V_t2, V_t1, V_m, V_a]: (p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2),s(t_h4s_nums_num,V_m)))) => p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),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_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_lbtrees_depthu_u_rulesu_c1, axiom, ![TV_u_27a]: ![V_x, V_t2, V_t1, V_m, V_a]: (p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_nums_num,V_m)))) => p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),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_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_lbtrees_depthu_u_rulesu_c0, axiom, ![TV_u_27a]: ![V_x, V_t2, V_t1]: p(s(t_bool,h4s_lbtrees_depth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))),s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_lbtrees_memu_u_cases, axiom, ![TV_u_27a]: ![V_a1, V_a0]: (p(s(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,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,h4s_lbtrees_mem(s(TV_u_27a,V_a0),s(t_h4s_lbtrees_lbtree(TV_u_27a),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_lbtrees_lbtreeu_u_strongu_u_bisimulation, axiom, ![TV_u_27a]: ![V_u, V_t]: (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u) <=> ?[V_R]: (p(s(t_bool,happ(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)),V_R),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u)))) & ![V_t0, V_u0]: (p(s(t_bool,happ(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)),V_R),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t0))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u0)))) => (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t0) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u0) | ?[V_a, V_t1, V_u1, V_t2, V_u2]: (p(s(t_bool,happ(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)),V_R),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u1)))) & (p(s(t_bool,happ(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)),V_R),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u2)))) & (s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t0) = 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),V_u0) = 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_u1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_u2))))))))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_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_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_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_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_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_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_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_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_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_SELECTu_u_REFL, axiom, ![TV_u_27a]: ![V_uu_0]: (![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_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ah4s_llists_llengthu_u_relu_u_def, axiom, ![TV_u_27a]: ![V_x, V_x0]: (p(s(t_bool,h4s_llists_llengthu_u_rel(s(t_h4s_llists_llist(TV_u_27a),V_x),s(t_h4s_nums_num,V_x0)))) <=> ![V_llengthu_u_relu_27]: (![V_a00, V_a10]: (((s(t_h4s_llists_llist(TV_u_27a),V_a00) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil) & s(t_h4s_nums_num,V_a10) = s(t_h4s_nums_num,h4s_nums_0)) | ?[V_h, V_n, 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))) & (s(t_h4s_nums_num,V_a10) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_llengthu_u_relu_27),s(t_h4s_llists_llist(TV_u_27a),V_t))),s(t_h4s_nums_num,V_n))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_llengthu_u_relu_27),s(t_h4s_llists_llist(TV_u_27a),V_a00))),s(t_h4s_nums_num,V_a10))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_llengthu_u_relu_27),s(t_h4s_llists_llist(TV_u_27a),V_x))),s(t_h4s_nums_num,V_x0))))))).
fof(ah4s_llists_LCONSu_u_NOTu_u_NILu_c0, axiom, ![TV_u_27a]: ![V_t, V_h]: ~ (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))).
fof(ah4s_llists_LLENGTH0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_ll, V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_llists_llengthu_u_rel(s(t_h4s_llists_llist(TV_u_27a),V_ll),s(t_h4s_nums_num,V_n))) => ![V_ll]: s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_mins_u_40(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))).
fof(ah4s_llists_LFINITEu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_bool,t)).
fof(ah4s_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_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_whiles_OLEASTu_u_INTRO, axiom, ![V_Q, V_P]: (((![V_n]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))) & ![V_n]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_n)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO, axiom, ![V_n]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_SOMEu_u_DEF, axiom, ![TV_u_27a]: ![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_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27a,V_x)))))).
fof(ah4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,t)) <=> s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_options_optionu_u_rep(s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_r))))) = s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_r))).
fof(ah4s_sums_INRu_u_INLu_u_11u_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
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_memu_u_def, axiom, ![TV_u_27a]: ![V_x, V_x0]: (p(s(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_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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_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_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_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_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_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_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_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_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_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_lbtrees_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_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_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_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_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_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_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_lbtreeu_u_absrepu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a))))) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a)).
fof(ah4s_lbtrees_lbtreeu_u_absrepu_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(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_isu_u_lbtreeu_u_def, axiom, ![TV_u_27a]: ![V_t]: (p(s(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_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_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_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_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_Ndrepu_u_def, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_v, V_t1, V_t2, V_xs]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_1),s(t_bool,V_v))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_h4s_lists_list(t_bool),V_xs))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1),s(t_h4s_lists_list(t_bool),V_xs))),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2),s(t_h4s_lists_list(t_bool),V_xs))))) => ![V_uu_0]: (![V_t1, V_t2, V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_bool,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_1),s(t_bool,V_v))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))) => ![V_t2, V_t1, V_a, V_x]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_ndrep(s(TV_u_27a,V_a),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_h4s_lists_list(t_bool),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_a))),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2)))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
fof(ah4s_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_lbtrees_Lfu_u_def, axiom, ![TV_u_27a]: s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lfrep)))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))))).
fof(ah4s_bools_DISJu_u_IMPu_u_THM, axiom, ![V_R, V_Q, V_P]: (((p(s(t_bool,V_P)) | p(s(t_bool,V_Q))) => p(s(t_bool,V_R))) <=> ((p(s(t_bool,V_P)) => p(s(t_bool,V_R))) & (p(s(t_bool,V_Q)) => p(s(t_bool,V_R)))))).
fof(ah4s_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_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_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_lbtrees_lbtreeu_u_caseu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: s(TV_u_27a,h4s_lbtrees_lbtreeu_u_case(s(TV_u_27a,V_e),s(t_fun(TV_u_27b,t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),t_fun(t_h4s_lbtrees_lbtree(TV_u_27b),TV_u_27a))),V_f),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_lf))) = s(TV_u_27a,V_e)).
fof(ah4s_bools_SELECTu_u_REFLu_u_2, axiom, ![TV_u_27a]: ![V_uu_0]: (![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_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ah4s_pairs_FORALLu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))) <=> ![V_pu_u_1, V_pu_u_2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
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_sats_dcu_u_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (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_s))))) & ((~ (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_s)) | ~ (p(s(t_bool,V_p))))))))))).
fof(ah4s_pairs_UNCURRYu_u_ONEu_u_ONEu_u_THM, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_g, V_f]: (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_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_g))) <=> s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_g))).
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_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(ch4s_lbtrees_mindepthu_u_thmu_c1, conjecture, ![TV_u_27a]: ![V_x, V_t2, V_t1, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_a)) & s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_mindepth(s(TV_u_27a,V_x),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_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_optionu_u_map(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_mindepth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))),s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_mindepth(s(TV_u_27a,V_x),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))))))))))).
