%   ORIGINAL: h4/inftree/relrec__rules_c0
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/sum/sum__Axiom: !g f. ?h. (!x. h (h4/sum/INL x) = f x) /\ (!y. h (h4/sum/INR y) = g y)
% Assm: h4/inftree/relrec__def: h4/inftree/relrec = (\a0 a1 a2 a3. !relrec_27. (!a00 a10 a20 a30. (?a. a20 = h4/inftree/iLf a /\ a30 = a00 a) \/ (?b df g. a20 = h4/inftree/iNd b df /\ a30 = a10 b g /\ (!d. relrec_27 a00 a10 (df d) (g d))) ==> relrec_27 a00 a10 a20 a30) ==> relrec_27 a0 a1 a2 a3)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pair/PAIR__MAP__THM: !y x g f. h4/pair/_23_23 f g (h4/pair/_2C x y) = h4/pair/_2C (f x) (g y)
% Assm: h4/sum/SUM__MAP__def_c1: !g f b. h4/sum/_2B_2B f g (h4/sum/INR b) = h4/sum/INR (g b)
% Assm: h4/sum/SUM__MAP__def_c0: !g f a. h4/sum/_2B_2B f g (h4/sum/INL a) = h4/sum/INL (f a)
% 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/inftree/iLf__def: !a. h4/inftree/iLf a = h4/inftree/to__inftree (\p. h4/sum/INL a)
% Assm: h4/inftree/inftree__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION h4/inftree/is__tree rep
% Assm: h4/inftree/inftree__11_c0: !a2 a1. h4/inftree/iLf a1 = h4/inftree/iLf a2 <=> a1 = a2
% Assm: h4/inftree/inftree__ind: !P. (!a. P (h4/inftree/iLf a)) /\ (!b f. (!d. P (f d)) ==> P (h4/inftree/iNd b f)) ==> (!t. P t)
% Assm: h4/inftree/inftree__distinct: !f b a. ~(h4/inftree/iLf a = h4/inftree/iNd b f)
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/PAIR__MAP: !p g f. h4/pair/_23_23 f g p = h4/pair/_2C (f (h4/pair/FST p)) (g (h4/pair/SND p))
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/list/MAP__ZIP: !l2 l1 g f. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> h4/list/MAP h4/pair/FST (h4/list/ZIP (h4/pair/_2C l1 l2)) = l1 /\ h4/list/MAP h4/pair/SND (h4/list/ZIP (h4/pair/_2C l1 l2)) = l2 /\ h4/list/MAP (h4/combin/o f h4/pair/FST) (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/list/MAP f l1 /\ h4/list/MAP (h4/combin/o g h4/pair/SND) (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/list/MAP g l2
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/inftree/inftree__bijections_c1: !r. h4/inftree/is__tree r <=> h4/inftree/from__inftree (h4/inftree/to__inftree r) = r
% Assm: h4/inftree/inftree__bijections_c0: !a. h4/inftree/to__inftree (h4/inftree/from__inftree a) = a
% 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/inftree/iNd__def: !f b. h4/inftree/iNd b f = h4/inftree/to__inftree (\p. h4/bool/COND (p = h4/list/NIL) (h4/sum/INR b) (h4/inftree/from__inftree (f (h4/list/HD p)) (h4/list/TL p)))
% Assm: h4/pair/C__UNCURRY__L: !x f. h4/combin/C (h4/pair/UNCURRY f) x = h4/pair/UNCURRY (h4/combin/C (h4/combin/o h4/combin/C f) x)
% Assm: h4/pair/LET2__RATOR: !b N M. h4/bool/LET (h4/pair/UNCURRY (\x y. N x y)) M b = h4/bool/LET (h4/pair/UNCURRY (\x y. N x y b)) M
% Assm: h4/pair/LET2__RAND: !P N M. P (h4/bool/LET (h4/pair/UNCURRY (\x y. N x y)) M) = h4/bool/LET (h4/pair/UNCURRY (\x y. P (N x y))) M
% Assm: h4/inftree/iNd__is__tree: !f b. h4/inftree/is__tree (\p. h4/bool/COND (p = h4/list/NIL) (h4/sum/INR b) (h4/inftree/from__inftree (f (h4/list/HD p)) (h4/list/TL p)))
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/inftree/inftree__11_c1: !f2 f1 b2 b1. h4/inftree/iNd b1 f1 = h4/inftree/iNd b2 f2 <=> b1 = b2 /\ f1 = f2
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% 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/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/inftree/is__tree__def: h4/inftree/is__tree = (\a0. !is__tree_27. (!a00. (?a. a00 = (\p. h4/sum/INL a)) \/ (?f b. a00 = (\p. h4/bool/COND (p = h4/list/NIL) (h4/sum/INR b) (f (h4/list/HD p) (h4/list/TL p))) /\ (!d. is__tree_27 (f d))) ==> is__tree_27 a00) ==> is__tree_27 a0)
% Assm: h4/bool/MONO__ALL: !Q P. (!x. P x ==> Q x) ==> (!x. P x) ==> (!x. Q x)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/pair/UNCURRY__VAR: !v f. h4/pair/UNCURRY f v = f (h4/pair/FST v) (h4/pair/SND v)
% 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/pair/UNCURRY0: !v f. h4/pair/UNCURRY f v = f (h4/pair/FST v) (h4/pair/SND v)
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/EQ__REFL: !x. x = 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/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/sum/sum__distinct: !y x. ~(h4/sum/INL x = h4/sum/INR y)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/list/ZIP0_c1: !x2 x1 l2 l1. h4/list/ZIP (h4/pair/_2C (h4/list/CONS x1 l1) (h4/list/CONS x2 l2)) = h4/list/CONS (h4/pair/_2C x1 x2) (h4/list/ZIP (h4/pair/_2C l1 l2))
% Assm: h4/list/ZIP0_c0: h4/list/ZIP (h4/pair/_2C h4/list/NIL h4/list/NIL) = h4/list/NIL
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/list__repfns_c1: !r. (\a0_27. !_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_27 a1) ==> _27list_27 a0_270) ==> _27list_27 a0_27) r <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/hidden____20__40ind____typelist0____def: h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% 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/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% 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/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/combin/K__DEF: h4/combin/K = (\x y. x)
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/list/NOT__NIL__CONS: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/list/TL0: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm: h4/list/HD0: !t h. h4/list/HD (h4/list/CONS h t) = h
% Assm: h4/sum/INR__INL__11_c1: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/list/LENGTH__CONS: !n l. h4/list/LENGTH l = h4/num/SUC n <=> (?h l_27. h4/list/LENGTH l_27 = n /\ l = h4/list/CONS h l_27)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/arithmetic/MAX__DEF: !n m. h4/arithmetic/MAX m n = h4/bool/COND (h4/prim__rec/_3C m n) n m
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/list/LAST__APPEND__CONS: !l2 l1 h. h4/list/LAST (h4/list/APPEND l1 (h4/list/CONS h l2)) = h4/list/LAST (h4/list/CONS h l2)
% Assm: h4/list/LAST__CONS__cond: !t h. h4/list/LAST (h4/list/CONS h t) = h4/bool/COND (t = h4/list/NIL) h (h4/list/LAST t)
% Assm: h4/list/APPEND__eq__NIL_c0: !l2 l1. h4/list/NIL = h4/list/APPEND l1 l2 <=> l1 = h4/list/NIL /\ l2 = h4/list/NIL
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/list/FOLDL__SNOC: !x l f e. h4/list/FOLDL f e (h4/list/SNOC x l) = f (h4/list/FOLDL f e l) x
% Goal: !nd lf a. h4/inftree/relrec lf nd (h4/inftree/iLf a) (lf a)
%   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_sums_sumu_u_Axiom]: !g f. ?h. (!x. happ h (h4/sum/INL x) = happ f x) /\ (!y. happ h (h4/sum/INR y) = happ g y)
% Assm [h4s_inftrees_relrecu_u_def]: !x x x x. h4/inftree/relrec x x x x <=> (!relrec_27. (!a00 a10 a20 a30. (?a. a20 = h4/inftree/iLf a /\ a30 = happ a00 a) \/ (?b df g. a20 = h4/inftree/iNd b df /\ a30 = happ (happ a10 b) g /\ (!d. happ (happ (happ (happ relrec_27 a00) a10) (happ df d)) (happ g d))) ==> happ (happ (happ (happ relrec_27 a00) a10) a20) a30) ==> happ (happ (happ (happ relrec_27 x) x) x) x)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_pairs_PAIRu_u_MAPu_u_THM]: !y x g f. h4/pair/_23_23 f g (h4/pair/_2C x y) = h4/pair/_2C (happ f x) (happ g y)
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c1]: !g f b. h4/sum/_2B_2B f g (h4/sum/INR b) = h4/sum/INR (happ g b)
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c0]: !g f a. h4/sum/_2B_2B f g (h4/sum/INL a) = h4/sum/INL (happ f a)
% 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_inftrees_iLfu_u_def]: !_0. (!a p. happ (happ _0 a) p = h4/sum/INL a) ==> (!a. h4/inftree/iLf a = h4/inftree/to__inftree (happ _0 a))
% Assm [h4s_inftrees_inftreeu_u_TYu_u_DEF]: ?rep. h4/bool/TYPE__DEFINITION h4/inftree/is__tree rep
% Assm [h4s_inftrees_inftreeu_u_11u_c0]: !a2 a1. h4/inftree/iLf a1 = h4/inftree/iLf a2 <=> a1 = a2
% Assm [h4s_inftrees_inftreeu_u_ind]: !P. (!a. happ P (h4/inftree/iLf a)) /\ (!b f. (!d. happ P (happ f d)) ==> happ P (h4/inftree/iNd b f)) ==> (!t. happ P t)
% Assm [h4s_inftrees_inftreeu_u_distinct]: !f b a. ~(h4/inftree/iLf a = h4/inftree/iNd b f)
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. happ h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_PAIRu_u_MAP]: !p g f. h4/pair/_23_23 f g p = h4/pair/_2C (happ f (happ h4/pair/FST p)) (happ g (happ h4/pair/SND p))
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_lists_MAPu_u_ZIP]: !l2 l1 g f. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> h4/list/MAP h4/pair/FST (h4/list/ZIP (h4/pair/_2C l1 l2)) = l1 /\ h4/list/MAP h4/pair/SND (h4/list/ZIP (h4/pair/_2C l1 l2)) = l2 /\ h4/list/MAP (h4/combin/o f h4/pair/FST) (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/list/MAP f l1 /\ h4/list/MAP (h4/combin/o g h4/pair/SND) (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/list/MAP g l2
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_inftrees_inftreeu_u_bijectionsu_c1]: !r. happ h4/inftree/is__tree r <=> h4/inftree/from__inftree (h4/inftree/to__inftree r) = r
% Assm [h4s_inftrees_inftreeu_u_bijectionsu_c0]: !a. h4/inftree/to__inftree (h4/inftree/from__inftree a) = a
% 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_inftrees_iNdu_u_def]: !_0. (!b f p. ?v. (v <=> p = h4/list/NIL) /\ happ (happ (happ _0 b) f) p = h4/bool/COND v (h4/sum/INR b) (happ (h4/inftree/from__inftree (happ f (h4/list/HD p))) (h4/list/TL p))) ==> (!f b. h4/inftree/iNd b f = h4/inftree/to__inftree (happ (happ _0 b) f))
% Assm [h4s_pairs_Cu_u_UNCURRYu_u_L]: !x f. happ (happ h4/combin/C (h4/pair/UNCURRY f)) x = h4/pair/UNCURRY (happ (happ h4/combin/C (h4/combin/o h4/combin/C f)) x)
% Assm [h4s_pairs_LET2u_u_RATOR]: !_3. (!N x b y. happ (happ (happ (happ _3 N) x) b) y = happ (happ (happ N x) y) b) ==> (!_2. (!N b x. happ (happ (happ _2 N) b) x = happ (happ (happ _3 N) x) b) ==> (!_1. (!N x y. happ (happ (happ _1 N) x) y = happ (happ N x) y) ==> (!_0. (!N x. happ (happ _0 N) x = happ (happ _1 N) x) ==> (!b N M. happ (h4/bool/LET (h4/pair/UNCURRY (happ _0 N)) M) b = h4/bool/LET (h4/pair/UNCURRY (happ (happ _2 N) b)) M))))
% Assm [h4s_pairs_LET2u_u_RAND]: !_3. (!P N x y. happ (happ (happ (happ _3 P) N) x) y = happ P (happ (happ N x) y)) ==> (!_2. (!P N x. happ (happ (happ _2 P) N) x = happ (happ (happ _3 P) N) x) ==> (!_1. (!N x y. happ (happ (happ _1 N) x) y = happ (happ N x) y) ==> (!_0. (!N x. happ (happ _0 N) x = happ (happ _1 N) x) ==> (!P N M. happ P (h4/bool/LET (h4/pair/UNCURRY (happ _0 N)) M) = h4/bool/LET (h4/pair/UNCURRY (happ (happ _2 P) N)) M))))
% Assm [h4s_inftrees_iNdu_u_isu_u_tree]: !_0. (!b f p. ?v. (v <=> p = h4/list/NIL) /\ happ (happ (happ _0 b) f) p = h4/bool/COND v (h4/sum/INR b) (happ (h4/inftree/from__inftree (happ f (h4/list/HD p))) (h4/list/TL p))) ==> (!f b. happ h4/inftree/is__tree (happ (happ _0 b) f))
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% 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_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_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_ETAu_u_AX]: !t x. happ t x = happ t 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_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_inftrees_inftreeu_u_11u_c1]: !f2 f1 b2 b1. h4/inftree/iNd b1 f1 = h4/inftree/iNd b2 f2 <=> b1 = b2 /\ f1 = f2
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% 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_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% 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_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_inftrees_isu_u_treeu_u_def]: !x. happ h4/inftree/is__tree x <=> (!is__tree_27. (!a00. (?a. !x. happ a00 x = h4/sum/INL a) \/ (?f b. (!x. ?v. (v <=> x = h4/list/NIL) /\ happ a00 x = h4/bool/COND v (h4/sum/INR b) (happ (happ f (h4/list/HD x)) (h4/list/TL x))) /\ (!d. happ is__tree_27 (happ f d))) ==> happ is__tree_27 a00) ==> happ is__tree_27 x)
% Assm [h4s_bools_MONOu_u_ALL]: !Q P. (!x. happ P x ==> happ Q x) ==> (!x. happ P x) ==> (!x. happ Q x)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_pairs_UNCURRYu_u_VAR]: !v f. happ (h4/pair/UNCURRY f) v = happ (happ f (happ h4/pair/FST v)) (happ h4/pair/SND v)
% 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_pairs_UNCURRY0]: !v f. happ (h4/pair/UNCURRY f) v = happ (happ f (happ h4/pair/FST v)) (happ h4/pair/SND v)
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (happ (happ h4/combin/C f) x) y = happ (happ f y) x
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = 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_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_sums_INRu_u_INLu_u_11u_c0]: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_sums_sumu_u_distinct]: !y x. ~(h4/sum/INL x = h4/sum/INR y)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_lists_ZIP0u_c1]: !x2 x1 l2 l1. h4/list/ZIP (h4/pair/_2C (happ (happ h4/list/CONS x1) l1) (happ (happ h4/list/CONS x2) l2)) = happ (happ h4/list/CONS (h4/pair/_2C x1 x2)) (h4/list/ZIP (h4/pair/_2C l1 l2))
% Assm [h4s_lists_ZIP0u_c0]: h4/list/ZIP (h4/pair/_2C h4/list/NIL h4/list/NIL) = h4/list/NIL
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (happ (happ h4/list/CONS h) t) = happ (happ h4/list/CONS (happ f h)) (h4/list/MAP f t)
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (happ (happ h4/list/CONS h) t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_listu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 r) <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r)
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% 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_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% 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_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Su_u_DEF]: !x x x. h4/combin/S x x x = happ (happ x x) (happ x x)
% Assm [h4s_combins_Ku_u_DEF]: !x x. h4/combin/K x x = x
% Assm [h4s_combins_Cu_u_DEF]: !x x x. happ (happ (happ h4/combin/C x) x) x = happ (happ x x) x
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_lists_NOTu_u_NILu_u_CONS]: !a1 a0. ~(h4/list/NIL = happ (happ h4/list/CONS a0) a1)
% Assm [h4s_lists_TL0]: !t h. h4/list/TL (happ (happ h4/list/CONS h) t) = t
% Assm [h4s_lists_HD0]: !t h. h4/list/HD (happ (happ h4/list/CONS h) t) = h
% Assm [h4s_sums_INRu_u_INLu_u_11u_c1]: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lists_LENGTHu_u_CONS]: !n l. h4/list/LENGTH l = h4/num/SUC n <=> (?h l_27. h4/list/LENGTH l_27 = n /\ l = happ (happ h4/list/CONS h) l_27)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (happ (happ h4/list/CONS h) t))) ==> (!l. happ P l)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. happ (happ h4/list/CONS a0) a1 = happ (happ h4/list/CONS a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_arithmetics_MAXu_u_DEF]: !n m. h4/arithmetic/MAX m n = h4/bool/COND (h4/prim__rec/_3C m n) n m
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_lists_LASTu_u_APPENDu_u_CONS]: !l2 l1 h. h4/list/LAST (h4/list/APPEND l1 (happ (happ h4/list/CONS h) l2)) = h4/list/LAST (happ (happ h4/list/CONS h) l2)
% Assm [h4s_lists_LASTu_u_CONSu_u_cond]: !t h. ?v. (v <=> t = h4/list/NIL) /\ h4/list/LAST (happ (happ h4/list/CONS h) t) = h4/bool/COND v h (h4/list/LAST t)
% Assm [h4s_lists_APPENDu_u_equ_u_NILu_c0]: !l2 l1. h4/list/NIL = h4/list/APPEND l1 l2 <=> l1 = h4/list/NIL /\ l2 = h4/list/NIL
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (happ (happ h4/list/CONS h) l1) l2 = happ (happ h4/list/CONS h) (h4/list/APPEND l1 l2)
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_lists_FOLDLu_u_SNOC]: !x l f e. h4/list/FOLDL f e (h4/list/SNOC x l) = happ (happ f (h4/list/FOLDL f e l)) x
% Goal: !nd lf a. h4/inftree/relrec lf nd (h4/inftree/iLf a) (happ lf a)
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_Q1393910,TV_Q1393906]: ![V_f, V_g]: (![V_x]: s(TV_Q1393906,happ(s(t_fun(TV_Q1393910,TV_Q1393906),V_f),s(TV_Q1393910,V_x))) = s(TV_Q1393906,happ(s(t_fun(TV_Q1393910,TV_Q1393906),V_g),s(TV_Q1393910,V_x))) => s(t_fun(TV_Q1393910,TV_Q1393906),V_f) = s(t_fun(TV_Q1393910,TV_Q1393906),V_g))).
fof(ah4s_sums_sumu_u_Axiom, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_g, V_f]: ?[V_h]: (![V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) & ![V_y]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(TV_u_27b,V_y))))).
fof(ah4s_inftrees_relrecu_u_def, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_x, V_x0, V_x1, V_x2]: (p(s(t_bool,h4s_inftrees_relrec(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_x0),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_x1),s(TV_u_27b,V_x2)))) <=> ![V_relrecu_27]: (![V_a00, V_a10, V_a20, V_a30]: ((?[V_a]: (s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_a20) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),h4s_inftrees_ilf(s(TV_u_27a,V_a))) & s(TV_u_27b,V_a30) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_a00),s(TV_u_27a,V_a)))) | ?[V_b, V_df, V_g]: (s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_a20) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),h4s_inftrees_ind(s(TV_u_27c,V_b),s(t_fun(TV_u_27d,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d)),V_df))) & (s(TV_u_27b,V_a30) = s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_a10),s(TV_u_27c,V_b))),s(t_fun(TV_u_27d,TV_u_27b),V_g))) & ![V_d]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_a00))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_a10))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27d,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d)),V_df),s(TV_u_27d,V_d))))),s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_g),s(TV_u_27d,V_d))))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_a00))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_a10))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_a20))),s(TV_u_27b,V_a30))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),t_fun(TV_u_27b,t_bool)))),V_relrecu_27),s(t_fun(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_x0))),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),V_x1))),s(TV_u_27b,V_x2))))))).
fof(ah4s_bools_TRUTH, axiom, p(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_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_pairs_PAIRu_u_MAPu_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_y, V_x, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27d),V_g),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27c,V_y))))) = s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_2c(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_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_g),s(TV_u_27c,V_y)))))).
fof(ah4s_sums_SUMu_u_MAPu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_g, V_f, V_b]: s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_u_2bu_2b(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_b))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_inr(s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,V_b)))))).
fof(ah4s_sums_SUMu_u_MAPu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27d,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_a]: s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_u_2bu_2b(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_inl(s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_a)))))).
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_inftrees_iLfu_u_def, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_a, V_p]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_uu_0),s(TV_u_27a,V_a))),s(t_h4s_lists_list(TV_u_27c),V_p))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a))) => ![V_a]: s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ilf(s(TV_u_27a,V_a))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_tou_u_inftree(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_uu_0),s(TV_u_27a,V_a))))))).
fof(ah4s_inftrees_inftreeu_u_TYu_u_DEF, axiom, ![TV_u_27d,TV_u_27a,TV_u_27b]: ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),h4s_inftrees_isu_u_tree),s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_rep))))).
fof(ah4s_inftrees_inftreeu_u_11u_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_a2, V_a1]: (s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ilf(s(TV_u_27a,V_a1))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ilf(s(TV_u_27a,V_a2))) <=> s(TV_u_27a,V_a1) = s(TV_u_27a,V_a2))).
fof(ah4s_inftrees_inftreeu_u_ind, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_P]: ((![V_a]: p(s(t_bool,happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),t_bool),V_P),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ilf(s(TV_u_27a,V_a)))))) & ![V_b, V_f]: (![V_d]: p(s(t_bool,happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),t_bool),V_P),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f),s(TV_u_27c,V_d)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),t_bool),V_P),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ind(s(TV_u_27b,V_b),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f)))))))) => ![V_t]: p(s(t_bool,happ(s(t_fun(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),t_bool),V_P),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),V_t)))))).
fof(ah4s_inftrees_inftreeu_u_distinct, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f, V_b, V_a]: ~ (s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ilf(s(TV_u_27a,V_a))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ind(s(TV_u_27b,V_b),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_PAIRu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))),s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_MAPu_u_ZIP, axiom, ![TV_u_27c,TV_u_27a,TV_u_27d,TV_u_27b]: ![V_l2, V_l1, V_g, V_f]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l2))) => (s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_lists_list(TV_u_27a),V_l1) & (s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_lists_list(TV_u_27b),V_l2) & (s(t_h4s_lists_list(TV_u_27c),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_lists_list(TV_u_27c),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_h4s_lists_list(TV_u_27a),V_l1))) & s(t_h4s_lists_list(TV_u_27d),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_lists_list(TV_u_27d),h4s_lists_map(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_lists_list(TV_u_27b),V_l2)))))))).
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_LETu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_inftrees_inftreeu_u_bijectionsu_c1, axiom, ![TV_u_27d,TV_u_27a,TV_u_27b]: ![V_r]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),h4s_inftrees_isu_u_tree),s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_r)))) <=> s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_inftrees_fromu_u_inftree(s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),h4s_inftrees_tou_u_inftree(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_r))))) = s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_r))).
fof(ah4s_inftrees_inftreeu_u_bijectionsu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27d]: ![V_a]: s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),h4s_inftrees_tou_u_inftree(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_inftrees_fromu_u_inftree(s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),V_a))))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27d),V_a)).
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_inftrees_iNdu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_b, V_f, V_p]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27c),V_p) = s(t_h4s_lists_list(TV_u_27c),h4s_lists_nil)) & s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27b,V_b))),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_lists_list(TV_u_27c),V_p))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_b))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_inftrees_fromu_u_inftree(s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f),s(TV_u_27c,h4s_lists_hd(s(t_h4s_lists_list(TV_u_27c),V_p))))))),s(t_h4s_lists_list(TV_u_27c),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27c),V_p)))))))) => ![V_f, V_b]: s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ind(s(TV_u_27b,V_b),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_tou_u_inftree(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27b,V_b))),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f))))))).
fof(ah4s_pairs_Cu_u_UNCURRYu_u_L, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_x, V_f]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(TV_u_27d,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),t_fun(TV_u_27d,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c))),h4s_combins_c),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),V_f))))),s(TV_u_27d,V_x))) = 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)),happ(s(t_fun(TV_u_27d,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27d,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27d,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)))),h4s_combins_c),s(t_fun(TV_u_27a,t_fun(TV_u_27d,t_fun(TV_u_27b,TV_u_27c))),h4s_combins_o(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c)),t_fun(TV_u_27d,t_fun(TV_u_27b,TV_u_27c))),h4s_combins_c),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),V_f))))),s(TV_u_27d,V_x)))))).
fof(ah4s_pairs_LET2u_u_RATOR, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a1,TV_u_27a2]: ![V_uu_3]: (![V_N, V_x, V_b, V_y]: s(TV_u_27c,happ(s(t_fun(TV_u_27a2,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c)))),V_uu_3),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))),s(TV_u_27b,V_b))),s(TV_u_27a2,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N),s(TV_u_27a1,V_x))),s(TV_u_27a2,V_y))),s(TV_u_27b,V_b))) => ![V_uu_2]: (![V_N, V_b, V_x]: s(t_fun(TV_u_27a2,TV_u_27c),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27b,t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c)))),V_uu_2),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27b,V_b))),s(TV_u_27a1,V_x))) = s(t_fun(TV_u_27a2,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c)))),V_uu_3),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))),s(TV_u_27b,V_b))) => ![V_uu_1]: (![V_N, V_x, V_y]: s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)))),V_uu_1),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))),s(TV_u_27a2,V_y))) = s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N),s(TV_u_27a1,V_x))),s(TV_u_27a2,V_y))) => ![V_uu_0]: (![V_N, V_x]: s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)))),V_uu_0),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))) = s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)))),V_uu_1),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))) => ![V_b, V_N, V_M]: s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),h4s_bools_let(s(t_fun(t_h4s_pairs_prod(TV_u_27a1,TV_u_27a2),t_fun(TV_u_27b,TV_u_27c)),h4s_pairs_uncurry(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)))),V_uu_0),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))))),s(t_h4s_pairs_prod(TV_u_27a1,TV_u_27a2),V_M))),s(TV_u_27b,V_b))) = s(TV_u_27c,h4s_bools_let(s(t_fun(t_h4s_pairs_prod(TV_u_27a1,TV_u_27a2),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27b,t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c)))),V_uu_2),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27b,V_b))))),s(t_h4s_pairs_prod(TV_u_27a1,TV_u_27a2),V_M)))))))).
fof(ah4s_pairs_LET2u_u_RAND, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_3]: (![V_P, V_N, V_x, V_y]: s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)))),V_uu_3),s(t_fun(TV_u_27c,TV_u_27d),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_P),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_N),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) => ![V_uu_2]: (![V_P, V_N, V_x]: s(t_fun(TV_u_27b,TV_u_27d),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)))),V_uu_2),s(t_fun(TV_u_27c,TV_u_27d),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,TV_u_27d),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)))),V_uu_3),s(t_fun(TV_u_27c,TV_u_27d),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) => ![V_uu_1]: (![V_N, V_x, 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),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_N),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) => ![V_uu_0]: (![V_N, V_x]: s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) => ![V_P, V_N, V_M]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_P),s(TV_u_27c,h4s_bools_let(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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_M))))) = s(TV_u_27d,h4s_bools_let(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)))),V_uu_2),s(t_fun(TV_u_27c,TV_u_27d),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_M)))))))).
fof(ah4s_inftrees_iNdu_u_isu_u_tree, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_b, V_f, V_p]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27b),V_p) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)) & s(t_h4s_sums_sum(TV_u_27c,TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_inftrees_inftree(TV_u_27c,TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_h4s_inftrees_inftree(TV_u_27c,TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27a)))),V_uu_0),s(TV_u_27a,V_b))),s(t_fun(TV_u_27b,t_h4s_inftrees_inftree(TV_u_27c,TV_u_27a,TV_u_27b)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_p))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27c,TV_u_27a),h4s_sums_inr(s(TV_u_27a,V_b))),s(t_h4s_sums_sum(TV_u_27c,TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27a)),h4s_inftrees_fromu_u_inftree(s(t_h4s_inftrees_inftree(TV_u_27c,TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_inftrees_inftree(TV_u_27c,TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,h4s_lists_hd(s(t_h4s_lists_list(TV_u_27b),V_p))))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27b),V_p)))))))) => ![V_f, V_b]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27a)),t_bool),h4s_inftrees_isu_u_tree),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_inftrees_inftree(TV_u_27c,TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_h4s_inftrees_inftree(TV_u_27c,TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27a)))),V_uu_0),s(TV_u_27a,V_b))),s(t_fun(TV_u_27b,t_h4s_inftrees_inftree(TV_u_27c,TV_u_27a,TV_u_27b)),V_f)))))))).
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_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_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_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_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_inftrees_inftreeu_u_11u_c1, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f2, V_f1, V_b2, V_b1]: (s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ind(s(TV_u_27b,V_b1),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f1))) = s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c),h4s_inftrees_ind(s(TV_u_27b,V_b2),s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f2))) <=> (s(TV_u_27b,V_b1) = s(TV_u_27b,V_b2) & s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f1) = s(t_fun(TV_u_27c,t_h4s_inftrees_inftree(TV_u_27a,TV_u_27b,TV_u_27c)),V_f2)))).
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_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_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_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_bools_LEFTu_u_ANDu_u_FORALLu_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_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_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_inftrees_isu_u_treeu_u_def, axiom, ![TV_u_27d,TV_u_27a,TV_u_27b]: ![V_x]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),h4s_inftrees_isu_u_tree),s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_x)))) <=> ![V_isu_u_treeu_27]: (![V_a00]: ((?[V_a]: ![V_x0]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_a00),s(t_h4s_lists_list(TV_u_27d),V_x0))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a))) | ?[V_f, V_b]: (![V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27d),V_x0) = s(t_h4s_lists_list(TV_u_27d),h4s_lists_nil)) & s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_a00),s(t_h4s_lists_list(TV_u_27d),V_x0))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_b))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27d,t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27d,h4s_lists_hd(s(t_h4s_lists_list(TV_u_27d),V_x0))))),s(t_h4s_lists_list(TV_u_27d),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27d),V_x0)))))))) & ![V_d]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),V_isu_u_treeu_27),s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27d,t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_f),s(TV_u_27d,V_d)))))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),V_isu_u_treeu_27),s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),V_isu_u_treeu_27),s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_x))))))).
fof(ah4s_bools_MONOu_u_ALL, 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_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_pairs_UNCURRYu_u_VAR, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_v, 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),V_v))) = 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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))))),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v)))))).
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_pairs_UNCURRY0, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_v, 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),V_v))) = 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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))))),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v)))))).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c))),h4s_combins_c),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(TV_u_27b,V_x))),s(TV_u_27a,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_y))),s(TV_u_27b,V_x)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_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_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_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_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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_sums_sumu_u_distinct, axiom, ![TV_u_27a,TV_u_27b]: ![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_inr(s(TV_u_27b,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_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_lists_ZIP0u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x2, V_x1, V_l2, V_l1]: s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x1))),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_x2))),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),h4s_lists_cons),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(TV_u_27b,V_x2))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2)))))))).
fof(ah4s_lists_ZIP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil)).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_listu_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_27listu_27]: (![V_a0u_270]: ((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,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_a0u_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,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_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_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(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_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_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_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(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,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_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(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_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_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_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(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,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
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_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
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_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_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
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_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_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,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_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_combins_Ku_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x),s(TV_u_27b,V_x0))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c))),h4s_combins_c),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x))),s(TV_u_27b,V_x0))),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_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_NOTu_u_NILu_u_CONS, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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))))).
fof(ah4s_lists_TL0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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),V_t)).
fof(ah4s_lists_HD0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(TV_u_27a,h4s_lists_hd(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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_h)).
fof(ah4s_sums_INRu_u_INLu_u_11u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_y))).
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_lists_LENGTHu_u_CONS, axiom, ![TV_u_27a]: ![V_n, V_l]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> ?[V_h, V_lu_27]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_lu_27))) = s(t_h4s_nums_num,V_n) & s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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_lu_27)))))).
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_lists_listu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0u_27))),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
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_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_arithmetics_MAXu_u_DEF, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_lists_LASTu_u_APPENDu_u_CONS, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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_l2))))))) = s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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_l2)))))).
fof(ah4s_lists_LASTu_u_CONSu_u_cond, axiom, ![TV_u_27a]: ![V_t, V_h]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27a),V_t) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)) & s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),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,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,V_h),s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_APPENDu_u_equ_u_NILu_c0, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))) <=> (s(t_h4s_lists_list(TV_u_27a),V_l1) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) & s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_lists_FOLDLu_u_SNOC, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),V_l))))),s(TV_u_27a,V_x)))).
fof(ch4s_inftrees_relrecu_u_rulesu_c0, conjecture, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_nd, V_lf, V_a]: p(s(t_bool,h4s_inftrees_relrec(s(t_fun(TV_u_27a,TV_u_27b),V_lf),s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27d,TV_u_27b),TV_u_27b)),V_nd),s(t_h4s_inftrees_inftree(TV_u_27a,TV_u_27c,TV_u_27d),h4s_inftrees_ilf(s(TV_u_27a,V_a))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_lf),s(TV_u_27a,V_a))))))).
