%   ORIGINAL: h4/Encode/lift__tree__def
% 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/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% 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/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> 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/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/sum/OUTL0: !x. h4/sum/OUTL (h4/sum/INL x) = x
% Assm: h4/sum/OUTR0: !x. h4/sum/OUTR (h4/sum/INR x) = x
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% 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/prim__rec/measure__def: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__add_c1: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__lte_c2: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/ISO0: !g f. h4/ind__type/ISO f g <=> (!x. f (g x) = x) /\ (!y. g (f y) = y)
% Assm: h4/ind__type/ISO__REFL: h4/ind__type/ISO (\x. x) (\x. x)
% Assm: h4/ind__type/ISO__FUN: !g_27 g f_27 f. h4/ind__type/ISO f f_27 /\ h4/ind__type/ISO g g_27 ==> h4/ind__type/ISO (\h a_27. g (h (f_27 a_27))) (\h a. g_27 (h (f a)))
% Assm: h4/ind__type/ISO__USAGE: !g f. h4/ind__type/ISO f g ==> (!P. (!x. P x) <=> (!x. P (g x))) /\ (!P. (?x. P x) <=> (?x. P (g x))) /\ (!a b. a = g b <=> f a = b)
% Assm: h4/list/list__Axiom: !f1 f0. ?fn. fn h4/list/NIL = f0 /\ (!a0 a1. fn (h4/list/CONS a0 a1) = f1 a0 a1 (fn a1))
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/MEM_c0: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm: h4/list/MEM_c1: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm: h4/list/EVERY__CONG: !l2 l1 P_27 P. l1 = l2 /\ (!x. h4/bool/IN x (h4/list/LIST__TO__SET l2) ==> (P x <=> P_27 x)) ==> (h4/list/EVERY P l1 <=> h4/list/EVERY P_27 l2)
% Assm: h4/Encode/tree__repfns_c0: !a. h4/Encode/_20_40ind__typeEncode4 (h4/Encode/_20_40ind__typeEncode5 a) = a
% Assm: h4/Encode/tree__repfns_c1: !r. (\a0_27. !_27tree_27 _27list_20_40ind__typeEncode0_27. (!a0_270. (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR h4/num/0 a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_20_40ind__typeEncode0_27 a1) ==> _27tree_27 a0_270) /\ (!a1_27. a1_27 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a1_27 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27tree_27 a0 /\ _27list_20_40ind__typeEncode0_27 a1) ==> _27list_20_40ind__typeEncode0_27 a1_27) ==> _27tree_27 a0_27) r <=> h4/Encode/_20_40ind__typeEncode5 (h4/Encode/_20_40ind__typeEncode4 r) = r
% Assm: h4/Encode/hidden__list__20__40ind____typeEncode0____repfns_c0: !a. h4/Encode/_20_40ind__typeEncode6 (h4/Encode/_20_40ind__typeEncode7 a) = a
% Assm: h4/Encode/hidden__list__20__40ind____typeEncode0____repfns_c1: !r. (\a1_27. !_27tree_27 _27list_20_40ind__typeEncode0_27. (!a0_27. (?a0 a1. a0_27 = (\a00 a10. h4/ind__type/CONSTR h4/num/0 a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_20_40ind__typeEncode0_27 a1) ==> _27tree_27 a0_27) /\ (!a1_270. a1_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a1_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27tree_27 a0 /\ _27list_20_40ind__typeEncode0_27 a1) ==> _27list_20_40ind__typeEncode0_27 a1_270) ==> _27list_20_40ind__typeEncode0_27 a1_27) r <=> h4/Encode/_20_40ind__typeEncode7 (h4/Encode/_20_40ind__typeEncode6 r) = r
% Assm: h4/Encode/hidden____20__40ind____typeEncode1____def: h4/Encode/_20_40ind__typeEncode1 = (\a0 a1. h4/Encode/_20_40ind__typeEncode4 ((\a00 a10. h4/ind__type/CONSTR h4/num/0 a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/Encode/_20_40ind__typeEncode7 a1)))
% Assm: h4/Encode/hidden____20__40ind____typeEncode2____def: h4/Encode/_20_40ind__typeEncode2 = h4/Encode/_20_40ind__typeEncode6 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/Encode/hidden____20__40ind____typeEncode3____def: h4/Encode/_20_40ind__typeEncode3 = (\a0 a1. h4/Encode/_20_40ind__typeEncode6 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/Encode/_20_40ind__typeEncode5 a0) (h4/Encode/_20_40ind__typeEncode7 a1)))
% Assm: h4/Encode/hidden____20__40ind____typeEncode8: h4/Encode/_20_40ind__typeEncode8 = (\a0 a1. h4/Encode/_20_40ind__typeEncode1 a0 (h4/min/_40 (\fn. fn h4/list/NIL = h4/Encode/_20_40ind__typeEncode2 /\ (!a00 a10. fn (h4/list/CONS a00 a10) = h4/Encode/_20_40ind__typeEncode3 a00 (fn a10))) a1))
% Assm: h4/Encode/Node0: h4/Encode/Node = h4/Encode/_20_40ind__typeEncode8
% Assm: h4/Encode/tree__case__def: !f a1 a0. h4/Encode/tree__CASE (h4/Encode/Node a0 a1) f = f a0 a1
% Assm: h4/Encode/tree__size__def_c0: !f a1 a0. h4/Encode/tree__size f (h4/Encode/Node a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (h4/Encode/tree1__size f a1))
% Assm: h4/Encode/tree__size__def_c2: !f a1 a0. h4/Encode/tree1__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/Encode/tree__size f a0) (h4/Encode/tree1__size f a1))
% Assm: h4/Encode/lift__tree__tupled__primitive__def: h4/Encode/lift__tree__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!a p ts a_27. h4/bool/IN a_27 (h4/list/LIST__TO__SET ts) ==> R (h4/pair/_2C p a_27) (h4/pair/_2C p (h4/Encode/Node a ts))))) (\lift__tree__tupled a_27. h4/pair/pair__CASE a_27 (\p v1. h4/Encode/tree__CASE v1 (\a ts. h4/combin/I (p a /\ h4/list/EVERY (\a0. lift__tree__tupled (h4/pair/_2C p a0)) ts))))
% Assm: h4/Encode/lift__tree__curried__def: !x1 x. h4/Encode/lift__tree x x1 <=> h4/Encode/lift__tree__tupled (h4/pair/_2C x x1)
% Goal: !ts p a. h4/Encode/lift__tree p (h4/Encode/Node a ts) <=> p a /\ h4/list/EVERY (h4/Encode/lift__tree p) ts
%   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_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% 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_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> 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_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_combins_ou_u_DEF]: !g f x. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_combins_Ku_u_THM]: !y x. h4/combin/K x y = x
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_sums_OUTL0]: !x. h4/sum/OUTL (h4/sum/INL x) = x
% Assm [h4s_sums_OUTR0]: !x. h4/sum/OUTR (h4/sum/INR x) = x
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (happ (h4/relation/inv__image R) f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_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_primu_u_recs_measureu_u_def]: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (happ h4/prim__rec/measure m)
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_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_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_addu_c1]: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_lteu_c2]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_ISO0]: !g f. h4/ind__type/ISO f g <=> (!x. happ f (happ g x) = x) /\ (!y. happ g (happ f y) = y)
% Assm [h4s_indu_u_types_ISOu_u_REFL]: !_0. (!x. happ _0 x = x) ==> h4/ind__type/ISO _0 _0
% Assm [h4s_indu_u_types_ISOu_u_FUN]: !_3. (!g_27 h f a. happ (happ (happ (happ _3 g_27) h) f) a = happ g_27 (happ h (happ f a))) ==> (!_2. (!g_27 f h. happ (happ (happ _2 g_27) f) h = happ (happ (happ _3 g_27) h) f) ==> (!_1. (!g h f_27 a_27. happ (happ (happ (happ _1 g) h) f_27) a_27 = happ g (happ h (happ f_27 a_27))) ==> (!_0. (!g f_27 h. happ (happ (happ _0 g) f_27) h = happ (happ (happ _1 g) h) f_27) ==> (!g_27 g f_27 f. h4/ind__type/ISO f f_27 /\ h4/ind__type/ISO g g_27 ==> h4/ind__type/ISO (happ (happ _0 g) f_27) (happ (happ _2 g_27) f)))))
% Assm [h4s_indu_u_types_ISOu_u_USAGE]: !g f. h4/ind__type/ISO f g ==> (!P. (!x. happ P x) <=> (!x. happ P (happ g x))) /\ (!P. (?x. happ P x) <=> (?x. happ P (happ g x))) /\ (!a b. a = happ g b <=> happ f a = b)
% Assm [h4s_lists_listu_u_Axiom]: !f1 f0. ?fn. happ fn h4/list/NIL = f0 /\ (!a0 a1. happ fn (h4/list/CONS a0 a1) = happ (happ (happ f1 a0) a1) (happ fn a1))
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_lists_MEMu_c0]: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm [h4s_lists_MEMu_c1]: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm [h4s_lists_EVERYu_u_CONG]: !l2 l1 P_27 P. l1 = l2 /\ (!x. h4/bool/IN x (h4/list/LIST__TO__SET l2) ==> (happ P x <=> happ P_27 x)) ==> (h4/list/EVERY P l1 <=> h4/list/EVERY P_27 l2)
% Assm [h4s_Encodes_treeu_u_repfnsu_c0]: !a. h4/Encode/_20_40ind__typeEncode4 (h4/Encode/_20_40ind__typeEncode5 a) = a
% Assm [h4s_Encodes_treeu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27tree_27 _27list_20_40ind__typeEncode0_27. (!a0_270. (?a0 a1. a0_270 = h4/ind__type/CONSTR h4/num/0 a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_20_40ind__typeEncode0_27 a1) ==> happ _27tree_27 a0_270) /\ (!a1_27. a1_27 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0 \/ (?a0 a1. a1_27 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27tree_27 a0 /\ happ _27list_20_40ind__typeEncode0_27 a1) ==> happ _27list_20_40ind__typeEncode0_27 a1_27) ==> happ _27tree_27 r) <=> h4/Encode/_20_40ind__typeEncode5 (h4/Encode/_20_40ind__typeEncode4 r) = r)
% Assm [h4s_Encodes_hiddenu_u_listu_u_20u_u_40indu_u_u_u_typeEncode0u_u_u_u_repfnsu_c0]: !a. h4/Encode/_20_40ind__typeEncode6 (h4/Encode/_20_40ind__typeEncode7 a) = a
% Assm [h4s_Encodes_hiddenu_u_listu_u_20u_u_40indu_u_u_u_typeEncode0u_u_u_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27tree_27 _27list_20_40ind__typeEncode0_27. (!a0_27. (?a0 a1. a0_27 = h4/ind__type/CONSTR h4/num/0 a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_20_40ind__typeEncode0_27 a1) ==> happ _27tree_27 a0_27) /\ (!a1_270. a1_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0 \/ (?a0 a1. a1_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27tree_27 a0 /\ happ _27list_20_40ind__typeEncode0_27 a1) ==> happ _27list_20_40ind__typeEncode0_27 a1_270) ==> happ _27list_20_40ind__typeEncode0_27 r) <=> h4/Encode/_20_40ind__typeEncode7 (h4/Encode/_20_40ind__typeEncode6 r) = r)
% Assm [h4s_Encodes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeEncode1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. h4/Encode/_20_40ind__typeEncode1 x x = h4/Encode/_20_40ind__typeEncode4 (h4/ind__type/CONSTR h4/num/0 x (h4/ind__type/FCONS (h4/Encode/_20_40ind__typeEncode7 x) _0)))
% Assm [h4s_Encodes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeEncode2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/Encode/_20_40ind__typeEncode2 = h4/Encode/_20_40ind__typeEncode6 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0)
% Assm [h4s_Encodes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeEncode3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. h4/Encode/_20_40ind__typeEncode3 x x = h4/Encode/_20_40ind__typeEncode6 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS (h4/Encode/_20_40ind__typeEncode5 x) (h4/ind__type/FCONS (h4/Encode/_20_40ind__typeEncode7 x) _0))))
% Assm [h4s_Encodes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeEncode8]: !_0. (!fn. happ _0 fn <=> happ fn h4/list/NIL = h4/Encode/_20_40ind__typeEncode2 /\ (!a00 a10. happ fn (h4/list/CONS a00 a10) = h4/Encode/_20_40ind__typeEncode3 a00 (happ fn a10))) ==> (!x x. happ (happ h4/Encode/_20_40ind__typeEncode8 x) x = h4/Encode/_20_40ind__typeEncode1 x (happ (h4/min/_40 _0) x))
% Assm [h4s_Encodes_Node0]: h4/Encode/Node = h4/Encode/_20_40ind__typeEncode8
% Assm [h4s_Encodes_treeu_u_caseu_u_def]: !f a1 a0. h4/Encode/tree__CASE (happ (happ h4/Encode/Node a0) a1) f = happ (happ f a0) a1
% Assm [h4s_Encodes_treeu_u_sizeu_u_defu_c0]: !f a1 a0. h4/Encode/tree__size f (happ (happ h4/Encode/Node a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (h4/Encode/tree1__size f a1))
% Assm [h4s_Encodes_treeu_u_sizeu_u_defu_c2]: !f a1 a0. h4/Encode/tree1__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/Encode/tree__size f a0) (h4/Encode/tree1__size f a1))
% Assm [h4s_Encodes_liftu_u_treeu_u_tupledu_u_primitiveu_u_def]: !_7. (!lift__tree__tupled p a0. happ (happ (happ _7 lift__tree__tupled) p) a0 <=> happ lift__tree__tupled (h4/pair/_2C p a0)) ==> (!_6. (!a lift__tree__tupled p ts. ?v. (v <=> happ p a /\ h4/list/EVERY (happ (happ _7 lift__tree__tupled) p) ts) /\ (happ (happ (happ (happ _6 a) lift__tree__tupled) p) ts <=> h4/combin/I v)) ==> (!_5. (!lift__tree__tupled p a. happ (happ (happ _5 lift__tree__tupled) p) a = happ (happ (happ _6 a) lift__tree__tupled) p) ==> (!_4. (!lift__tree__tupled p v1. happ (happ (happ _4 lift__tree__tupled) p) v1 <=> h4/Encode/tree__CASE v1 (happ (happ _5 lift__tree__tupled) p)) ==> (!_3. (!lift__tree__tupled p. happ (happ _3 lift__tree__tupled) p = happ (happ _4 lift__tree__tupled) p) ==> (!_2. (!lift__tree__tupled a_27. happ (happ _2 lift__tree__tupled) a_27 <=> h4/pair/pair__CASE a_27 (happ _3 lift__tree__tupled)) ==> (!_1. (!lift__tree__tupled. happ _1 lift__tree__tupled = happ _2 lift__tree__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!a p ts a_27. h4/bool/IN a_27 (h4/list/LIST__TO__SET ts) ==> happ (happ R (h4/pair/_2C p a_27)) (h4/pair/_2C p (happ (happ h4/Encode/Node a) ts)))) ==> h4/Encode/lift__tree__tupled = h4/relation/WFREC (h4/min/_40 _0) _1)))))))
% Assm [h4s_Encodes_liftu_u_treeu_u_curriedu_u_def]: !x1 x. happ (h4/Encode/lift__tree x) x1 <=> happ h4/Encode/lift__tree__tupled (h4/pair/_2C x x1)
% Goal: !ts p a. happ (h4/Encode/lift__tree p) (happ (happ h4/Encode/Node a) ts) <=> happ p a /\ h4/list/EVERY (h4/Encode/lift__tree p) ts
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_Q204638,TV_Q204634]: ![V_f, V_g]: (![V_x]: s(TV_Q204634,happ(s(t_fun(TV_Q204638,TV_Q204634),V_f),s(TV_Q204638,V_x))) = s(TV_Q204634,happ(s(t_fun(TV_Q204638,TV_Q204634),V_g),s(TV_Q204638,V_x))) => s(t_fun(TV_Q204638,TV_Q204634),V_f) = s(t_fun(TV_Q204638,TV_Q204634),V_g))).
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_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_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_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_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_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_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_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_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_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_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_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_sums_OUTL0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_sums_outl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_sums_OUTR0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(TV_u_27b,h4s_sums_outr(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_x))))) = s(TV_u_27b,V_x)).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_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_primu_u_recs_measureu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure) = s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c)))).
fof(ah4s_primu_u_recs_WFu_u_measure, axiom, ![TV_u_27a]: ![V_m]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure),s(t_fun(TV_u_27a,t_h4s_nums_num),V_m))))))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_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_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_addu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_lteu_c2, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_ISO0, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,TV_u_27a),V_g)))) <=> (![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))) = s(TV_u_27b,V_x) & ![V_y]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))) = s(TV_u_27a,V_y)))).
fof(ah4s_indu_u_types_ISOu_u_REFL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_uu_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x) => p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27a),V_uu_0),s(t_fun(TV_u_27a,TV_u_27a),V_uu_0)))))).
fof(ah4s_indu_u_types_ISOu_u_FUN, axiom, ![TV_u_27d,TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_3]: (![V_gu_27, V_h, V_f, V_a]: 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_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_3),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27c,TV_u_27d),V_h))),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(TV_u_27a,V_a))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_gu_27),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_a))))))) => ![V_uu_2]: (![V_gu_27, V_f, V_h]: s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27c,TV_u_27d),V_h))) = s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_3),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27c,TV_u_27d),V_h))),s(t_fun(TV_u_27a,TV_u_27c),V_f))) => ![V_uu_1]: (![V_g, V_h, V_fu_27, V_au_27]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)))),V_uu_1),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27a,TV_u_27b),V_h))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(TV_u_27c,V_au_27))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_h),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_fu_27),s(TV_u_27c,V_au_27))))))) => ![V_uu_0]: (![V_g, V_fu_27, V_h]: s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)))),V_uu_0),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(t_fun(TV_u_27a,TV_u_27b),V_h))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)))),V_uu_1),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27a,TV_u_27b),V_h))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))) => ![V_gu_27, V_g, V_fu_27, V_f]: ((p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27)))) & p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))))) => p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)))),V_uu_0),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27a,TV_u_27c),V_f)))))))))))).
fof(ah4s_indu_u_types_ISOu_u_USAGE, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,TV_u_27a),V_g)))) => (![V_P]: (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) <=> ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))))) & (![V_P]: (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) <=> ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))))) & ![V_a, V_b]: (s(TV_u_27a,V_a) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_b))) <=> s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a))) = s(TV_u_27b,V_b)))))).
fof(ah4s_lists_listu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f1, V_f0]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_f0) & ![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b))),V_f1),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),V_a1))))))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_MEMu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))) = s(t_bool,f)).
fof(ah4s_lists_MEMu_c1, axiom, ![TV_u_27a]: ![V_x, V_t, V_h]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_h) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_t))))))))).
fof(ah4s_lists_EVERYu_u_CONG, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_Pu_27, V_P]: ((s(t_h4s_lists_list(TV_u_27a),V_l1) = s(t_h4s_lists_list(TV_u_27a),V_l2) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l2)))))) => s(t_bool,happ(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_Pu_27),s(TV_u_27a,V_x))))) => s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_Pu_27),s(t_h4s_lists_list(TV_u_27a),V_l2))))).
fof(ah4s_Encodes_treeu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_Encodes_tree(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode5(s(t_h4s_Encodes_tree(TV_u_27a),V_a))))) = s(t_h4s_Encodes_tree(TV_u_27a),V_a)).
fof(ah4s_Encodes_treeu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27treeu_27, V_uu_27listu_20u_40indu_u_typeEncode0u_27]: ((![V_a0u_270]: (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_20u_40indu_u_typeEncode0u_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27treeu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) & ![V_a1u_27]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1u_27) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1u_27) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27treeu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_20u_40indu_u_typeEncode0u_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_20u_40indu_u_typeEncode0u_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1u_27)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27treeu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode5(s(t_h4s_Encodes_tree(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode4(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_Encodes_hiddenu_u_listu_u_20u_u_40indu_u_u_u_typeEncode0u_u_u_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode6(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode7(s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),V_a))))) = s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),V_a)).
fof(ah4s_Encodes_hiddenu_u_listu_u_20u_u_40indu_u_u_u_typeEncode0u_u_u_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27treeu_27, V_uu_27listu_20u_40indu_u_typeEncode0u_27]: ((![V_a0u_27]: (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_27) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_20u_40indu_u_typeEncode0u_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27treeu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_27))))) & ![V_a1u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27treeu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_20u_40indu_u_typeEncode0u_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_20u_40indu_u_typeEncode0u_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1u_270)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_20u_40indu_u_typeEncode0u_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode7(s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode6(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_Encodes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeEncode1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_Encodes_tree(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode1(s(TV_u_27a,V_x),s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),V_x0))) = s(t_h4s_Encodes_tree(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode7(s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_Encodes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeEncode2u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode2) = s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode6(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_Encodes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeEncode3u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode3(s(t_h4s_Encodes_tree(TV_u_27a),V_x),s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),V_x0))) = s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode6(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode5(s(t_h4s_Encodes_tree(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode7(s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))))).
fof(ah4s_Encodes_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeEncode8, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_fn]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a)),t_bool),V_uu_0),s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a)),V_fn)))) <=> (s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a)),V_fn),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),h4s_lists_nil))) = s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode2) & ![V_a00, V_a10]: s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a)),V_fn),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),h4s_lists_cons(s(t_h4s_Encodes_tree(TV_u_27a),V_a00),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_a10))))) = s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode3(s(t_h4s_Encodes_tree(TV_u_27a),V_a00),s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a)),V_fn),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_a10))))))) => ![V_x, V_x0]: s(t_h4s_Encodes_tree(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a))),h4s_encodes_u_20u_40indu_u_typeencode8),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_x0))) = s(t_h4s_Encodes_tree(TV_u_27a),h4s_encodes_u_20u_40indu_u_typeencode1(s(TV_u_27a,V_x),s(t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_listu_20u_40indu_u_typeEncode0(TV_u_27a)),t_bool),V_uu_0))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_x0))))))).
fof(ah4s_Encodes_Node0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a))),h4s_encodes_node) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a))),h4s_encodes_u_20u_40indu_u_typeencode8)).
fof(ah4s_Encodes_treeu_u_caseu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_a1, V_a0]: s(TV_u_27b,h4s_encodes_treeu_u_case(s(t_h4s_Encodes_tree(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a))),h4s_encodes_node),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_a1))),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_a1)))).
fof(ah4s_Encodes_treeu_u_sizeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_encodes_treeu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_Encodes_tree(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a))),h4s_encodes_node),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,h4s_encodes_tree1u_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_a1)))))))).
fof(ah4s_Encodes_treeu_u_sizeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_encodes_tree1u_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),h4s_lists_cons(s(t_h4s_Encodes_tree(TV_u_27a),V_a0),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_encodes_treeu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_Encodes_tree(TV_u_27a),V_a0))),s(t_h4s_nums_num,h4s_encodes_tree1u_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_a1)))))))).
fof(ah4s_Encodes_liftu_u_treeu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_7]: (![V_liftu_u_treeu_u_tupled, V_p, V_a0]: s(t_bool,happ(s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_Encodes_tree(TV_u_27a),V_a0))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_Encodes_tree(TV_u_27a),V_a0))))) => ![V_uu_6]: (![V_a, V_liftu_u_treeu_u_tupled, V_p, V_ts]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_a)))) & p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_ts)))))) & s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool)))),V_uu_6),s(TV_u_27a,V_a))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_ts))) = s(t_bool,h4s_combins_i(s(t_bool,V_v)))) => ![V_uu_5]: (![V_liftu_u_treeu_u_tupled, V_p, V_a]: s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))),s(TV_u_27a,V_a))) = s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool)))),V_uu_6),s(TV_u_27a,V_a))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))) => ![V_uu_4]: (![V_liftu_u_treeu_u_tupled, V_p, V_v1]: s(t_bool,happ(s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_Encodes_tree(TV_u_27a),V_v1))) = s(t_bool,h4s_encodes_treeu_u_case(s(t_h4s_Encodes_tree(TV_u_27a),V_v1),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_bool)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))))) => ![V_uu_3]: (![V_liftu_u_treeu_u_tupled, V_p]: s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))) = s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_fun(TV_u_27a,t_bool),V_p))) => ![V_uu_2]: (![V_liftu_u_treeu_u_tupled, V_au_27]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),V_au_27))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),V_au_27),s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))))) => ![V_uu_1]: (![V_liftu_u_treeu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),V_liftu_u_treeu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),V_R)))) & ![V_a, V_p, V_ts, V_au_27]: (p(s(t_bool,h4s_bools_in(s(t_h4s_Encodes_tree(TV_u_27a),V_au_27),s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_ts)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_Encodes_tree(TV_u_27a),V_au_27))))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_Encodes_tree(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a))),h4s_encodes_node),s(TV_u_27a,V_a))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_ts))))))))))) => s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),h4s_encodes_liftu_u_treeu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool)),V_uu_1)))))))))))).
fof(ah4s_Encodes_liftu_u_treeu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x1, V_x]: s(t_bool,happ(s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),h4s_encodes_liftu_u_tree(s(t_fun(TV_u_27a,t_bool),V_x))),s(t_h4s_Encodes_tree(TV_u_27a),V_x1))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),t_bool),h4s_encodes_liftu_u_treeu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_Encodes_tree(TV_u_27a)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(t_h4s_Encodes_tree(TV_u_27a),V_x1)))))).
fof(ch4s_Encodes_liftu_u_treeu_u_def, conjecture, ![TV_u_27a]: ![V_ts, V_p, V_a]: (p(s(t_bool,happ(s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),h4s_encodes_liftu_u_tree(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_Encodes_tree(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),t_h4s_Encodes_tree(TV_u_27a))),h4s_encodes_node),s(TV_u_27a,V_a))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_ts)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_a)))) & p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),h4s_encodes_liftu_u_tree(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_lists_list(t_h4s_Encodes_tree(TV_u_27a)),V_ts))))))).
