%   ORIGINAL: h4/state__transformer/JOIN__MAP__JOIN
% 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/state__transformer/JOIN__DEF: !z. h4/state__transformer/JOIN z = h4/state__transformer/BIND z h4/combin/I
% Assm: h4/state__transformer/MMAP__JOIN: !f. h4/combin/o (h4/state__transformer/MMAP f) h4/state__transformer/JOIN = h4/combin/o h4/state__transformer/JOIN (h4/state__transformer/MMAP (h4/state__transformer/MMAP f))
% Assm: h4/state__transformer/JOIN__MMAP__UNIT: h4/combin/o h4/state__transformer/JOIN (h4/state__transformer/MMAP h4/state__transformer/UNIT) = h4/combin/I
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/state__transformer/MMAP__DEF: !m f. h4/state__transformer/MMAP f m = h4/state__transformer/BIND m (h4/combin/o h4/state__transformer/UNIT f)
% Assm: h4/state__transformer/BIND__LEFT__UNIT: !x k. h4/state__transformer/BIND (h4/state__transformer/UNIT x) k = k x
% Assm: h4/state__transformer/JOIN__UNIT: h4/combin/o h4/state__transformer/JOIN h4/state__transformer/UNIT = h4/combin/I
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__EXT: !g f. (!x. f x = g x) ==> f = g
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/state__transformer/BIND__ASSOC: !n m k. h4/state__transformer/BIND k (\a. h4/state__transformer/BIND (m a) n) = h4/state__transformer/BIND (h4/state__transformer/BIND k m) n
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/state__transformer/BIND__RIGHT__UNIT: !k. h4/state__transformer/BIND k h4/state__transformer/UNIT = k
% 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/I__DEF: h4/combin/I = h4/combin/S h4/combin/K h4/combin/K
% Assm: h4/state__transformer/MMAP__COMP: !g f. h4/state__transformer/MMAP (h4/combin/o f g) = h4/combin/o (h4/state__transformer/MMAP f) (h4/state__transformer/MMAP g)
% Assm: h4/state__transformer/MMAP__UNIT: !f. h4/combin/o (h4/state__transformer/MMAP f) h4/state__transformer/UNIT = h4/combin/o h4/state__transformer/UNIT f
% Assm: h4/state__transformer/MMAP__ID: h4/state__transformer/MMAP h4/combin/I = h4/combin/I
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/list/LIST__APPLY__o: !xs gs fs. h4/list/LIST__APPLY (h4/list/LIST__APPLY (h4/list/LIST__APPLY (h4/list/CONS h4/combin/o h4/list/NIL) fs) gs) xs = h4/list/LIST__APPLY fs (h4/list/LIST__APPLY gs xs)
% Assm: h4/option/OPTION__APPLY__o: !x g f. h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/SOME h4/combin/o) f) g) x = h4/option/OPTION__APPLY f (h4/option/OPTION__APPLY g x)
% Assm: h4/pair/UNCURRY0: !v f. h4/pair/UNCURRY f v = f (h4/pair/FST v) (h4/pair/SND v)
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% 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/arithmetic/DIVMOD__DEF: h4/arithmetic/DIVMOD = h4/relation/WFREC (h4/prim__rec/measure (h4/combin/o h4/pair/FST h4/pair/SND)) (\f. h4/pair/UNCURRY (\a. h4/pair/UNCURRY (\m n. h4/bool/COND (n = h4/num/0) (h4/pair/_2C h4/num/0 h4/num/0) (h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (\q. f (h4/pair/_2C (h4/arithmetic/_2B a q) (h4/pair/_2C (h4/arithmetic/_2D m (h4/arithmetic/_2A n q)) n))) (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n))))))))
% 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/FORALL__UNCURRY: !f. $forall (h4/pair/UNCURRY f) <=> $forall (h4/combin/o $forall f)
% Assm: h4/state__transformer/BIND__DEF: !g f. h4/state__transformer/BIND g f = h4/combin/o (h4/pair/UNCURRY f) g
% Assm: h4/state__transformer/mapM__def: !f. h4/state__transformer/mapM f = h4/combin/o h4/state__transformer/sequence (h4/list/MAP f)
% Assm: h4/pair/o__UNCURRY__R: !g f. h4/combin/o f (h4/pair/UNCURRY g) = h4/pair/UNCURRY (h4/combin/o (h4/combin/o f) g)
% Assm: h4/pair/S__UNCURRY__R: !g f. h4/combin/S f (h4/pair/UNCURRY g) = h4/pair/UNCURRY (h4/combin/S (h4/combin/o h4/combin/S (h4/combin/o (h4/combin/o f) h4/pair/_2C)) g)
% Assm: h4/list/TL__GENLIST: !n f. h4/list/TL (h4/list/GENLIST f (h4/num/SUC n)) = h4/list/GENLIST (h4/combin/o f h4/num/SUC) n
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/sum/SUM__MAP__CASE: !z g f. h4/sum/_2B_2B f g z = h4/sum/sum__CASE z (h4/combin/o h4/sum/INL f) (h4/combin/o h4/sum/INR g)
% Assm: h4/option/OPTION__APPLY__def_c1: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__APPLY__def_c0: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/list/splitAtPki__DEF_c1: !t k h P. h4/list/splitAtPki P k (h4/list/CONS h t) = h4/bool/COND (P h4/num/0 h) (k h4/list/NIL (h4/list/CONS h t)) (h4/list/splitAtPki (h4/combin/o P h4/num/SUC) (\p s. k (h4/list/CONS h p) s) t)
% Assm: h4/combin/GEN__LET__RAND: !v f P. P (h4/bool/LET f v) = h4/bool/LET (h4/combin/o P f) v
% Assm: h4/list/DROP__splitAtPki: !n l. h4/list/DROP n l = h4/list/splitAtPki (h4/combin/o h4/combin/K ($equals n)) (h4/combin/K h4/combin/I) l
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% 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/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/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/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% 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/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/EQ__CLAUSES_c1: !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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/list/MAP__LIST__BIND: !l g f. h4/list/MAP f (h4/list/LIST__BIND l g) = h4/list/LIST__BIND l (h4/combin/o (h4/list/MAP f) g)
% Assm: h4/list/LIST__APPLY__DEF: !xs fs. h4/list/LIST__APPLY fs xs = h4/list/LIST__BIND fs (h4/combin/C h4/list/MAP xs)
% Assm: h4/list/SINGL__LIST__APPLY__L: !x f. h4/list/LIST__BIND (h4/list/CONS x h4/list/NIL) f = f x
% Assm: h4/list/LIST__BIND__MAP: !l g f. h4/list/LIST__BIND (h4/list/MAP f l) g = h4/list/LIST__BIND l (h4/combin/o g f)
% Assm: h4/list/LIST__BIND__APPEND: !l2 l1 f. h4/list/LIST__BIND (h4/list/APPEND l1 l2) f = h4/list/APPEND (h4/list/LIST__BIND l1 f) (h4/list/LIST__BIND l2 f)
% Assm: h4/list/LIST__BIND__THM_c0: !f. h4/list/LIST__BIND h4/list/NIL f = h4/list/NIL
% Assm: h4/list/LIST__BIND__THM_c1: !t h f. h4/list/LIST__BIND (h4/list/CONS h t) f = h4/list/APPEND (f h) (h4/list/LIST__BIND t f)
% Assm: h4/list/APPEND__11_c1: !l3 l2 l1. h4/list/APPEND l2 l1 = h4/list/APPEND l3 l1 <=> l2 = l3
% Assm: h4/list/MAP__MAP__o: !l g f. h4/list/MAP f (h4/list/MAP g l) = h4/list/MAP (h4/combin/o f g) l
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% 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/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/pair/FORALL__PROD: !P. (!p. P p) <=> (!p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/combin/S__THM: !x g f. h4/combin/S f g x = f x (g x)
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/list/ZIP0_c0: h4/list/ZIP (h4/pair/_2C h4/list/NIL 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/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/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/FALSITY: !t. F ==> t
% Assm: h4/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = 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/sum/INR__INL__11_c1: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/sum/FORALL__SUM: !P. (!s. P s) <=> (!x. P (h4/sum/INL x)) /\ (!y. P (h4/sum/INR y))
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/list/EL__GENLIST: !x n f. h4/prim__rec/_3C x n ==> h4/list/EL x (h4/list/GENLIST f n) = f x
% Assm: h4/list/LENGTH__GENLIST: !n f. h4/list/LENGTH (h4/list/GENLIST f n) = n
% Assm: h4/list/LIST__EQ: !l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ (!x. h4/prim__rec/_3C x (h4/list/LENGTH l1) ==> h4/list/EL x l1 = h4/list/EL x l2) ==> l1 = l2
% Assm: h4/list/LENGTH__TL: !l. h4/prim__rec/_3C h4/num/0 (h4/list/LENGTH l) ==> h4/list/LENGTH (h4/list/TL l) = h4/arithmetic/_2D (h4/list/LENGTH l) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/list/EL0_c1: !n l. h4/list/EL (h4/num/SUC n) l = h4/list/EL n (h4/list/TL l)
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/arithmetic/SUC__SUB1: !m. h4/arithmetic/_2D (h4/num/SUC m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/list/DROP__LENGTH__TOO__LONG: !n l. h4/arithmetic/_3C_3D (h4/list/LENGTH l) n ==> h4/list/DROP n l = h4/list/NIL
% Assm: h4/list/splitAtPki__EQN: !l k P. h4/list/splitAtPki P k l = h4/option/option__CASE (h4/while/OLEAST (\i. h4/prim__rec/_3C i (h4/list/LENGTH l) /\ P i (h4/list/EL i l))) (k l h4/list/NIL) (\i. k (h4/list/TAKE i l) (h4/list/DROP i l))
% Assm: h4/while/OLEAST__INTRO: !Q P. ((!n. ~P n) ==> Q h4/option/NONE) /\ (!n. P n /\ (!m. h4/prim__rec/_3C m n ==> ~P m) ==> Q (h4/option/SOME n)) ==> Q (h4/while/OLEAST P)
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Goal: h4/combin/o h4/state__transformer/JOIN (h4/state__transformer/MMAP h4/state__transformer/JOIN) = h4/combin/o h4/state__transformer/JOIN h4/state__transformer/JOIN
%   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_stateu_u_transformers_JOINu_u_DEF]: !z. happ h4/state__transformer/JOIN z = h4/state__transformer/BIND z h4/combin/I
% Assm [h4s_stateu_u_transformers_MMAPu_u_JOIN]: !f. happ (happ h4/combin/o (h4/state__transformer/MMAP f)) h4/state__transformer/JOIN = happ (happ h4/combin/o h4/state__transformer/JOIN) (h4/state__transformer/MMAP (h4/state__transformer/MMAP f))
% Assm [h4s_stateu_u_transformers_JOINu_u_MMAPu_u_UNIT]: happ (happ h4/combin/o h4/state__transformer/JOIN) (h4/state__transformer/MMAP h4/state__transformer/UNIT) = h4/combin/I
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (happ (happ h4/combin/o f) g) x = happ f (happ g x)
% Assm [h4s_stateu_u_transformers_MMAPu_u_DEF]: !m f. happ (h4/state__transformer/MMAP f) m = h4/state__transformer/BIND m (happ (happ h4/combin/o h4/state__transformer/UNIT) f)
% Assm [h4s_stateu_u_transformers_BINDu_u_LEFTu_u_UNIT]: !x k. h4/state__transformer/BIND (happ h4/state__transformer/UNIT x) k = happ k x
% Assm [h4s_stateu_u_transformers_JOINu_u_UNIT]: happ (happ h4/combin/o h4/state__transformer/JOIN) h4/state__transformer/UNIT = h4/combin/I
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_EXT]: !g f. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_stateu_u_transformers_BINDu_u_ASSOC]: !_0. (!m n a. happ (happ (happ _0 m) n) a = h4/state__transformer/BIND (happ m a) n) ==> (!n m k. h4/state__transformer/BIND k (happ (happ _0 m) n) = h4/state__transformer/BIND (h4/state__transformer/BIND k m) n)
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_stateu_u_transformers_BINDu_u_RIGHTu_u_UNIT]: !k. h4/state__transformer/BIND k h4/state__transformer/UNIT = k
% Assm [h4s_combins_Su_u_DEF]: !x x x. happ (happ (happ h4/combin/S x) x) x = happ (happ x x) (happ x x)
% Assm [h4s_combins_Ku_u_DEF]: !x x. happ (happ h4/combin/K x) x = x
% Assm [h4s_combins_Iu_u_DEF]: h4/combin/I = happ (happ h4/combin/S h4/combin/K) h4/combin/K
% Assm [h4s_stateu_u_transformers_MMAPu_u_COMP]: !g f. h4/state__transformer/MMAP (happ (happ h4/combin/o f) g) = happ (happ h4/combin/o (h4/state__transformer/MMAP f)) (h4/state__transformer/MMAP g)
% Assm [h4s_stateu_u_transformers_MMAPu_u_UNIT]: !f. happ (happ h4/combin/o (h4/state__transformer/MMAP f)) h4/state__transformer/UNIT = happ (happ h4/combin/o h4/state__transformer/UNIT) f
% Assm [h4s_stateu_u_transformers_MMAPu_u_ID]: h4/state__transformer/MMAP h4/combin/I = h4/combin/I
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (happ (happ h4/combin/o f) g) x = happ f (happ g x)
% Assm [h4s_lists_LISTu_u_APPLYu_u_o]: !xs gs fs. h4/list/LIST__APPLY (h4/list/LIST__APPLY (h4/list/LIST__APPLY (happ (happ h4/list/CONS h4/combin/o) h4/list/NIL) fs) gs) xs = h4/list/LIST__APPLY fs (h4/list/LIST__APPLY gs xs)
% Assm [h4s_options_OPTIONu_u_APPLYu_u_o]: !x g f. h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/OPTION__APPLY (h4/option/SOME h4/combin/o) f) g) x = h4/option/OPTION__APPLY f (h4/option/OPTION__APPLY g x)
% 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_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_lists_MAPu_u_ZIP]: !l2 l1 g f. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> happ (happ h4/list/MAP h4/pair/FST) (h4/list/ZIP (happ (happ h4/pair/_2C l1) l2)) = l1 /\ happ (happ h4/list/MAP h4/pair/SND) (h4/list/ZIP (happ (happ h4/pair/_2C l1) l2)) = l2 /\ happ (happ h4/list/MAP (happ (happ h4/combin/o f) h4/pair/FST)) (h4/list/ZIP (happ (happ h4/pair/_2C l1) l2)) = happ (happ h4/list/MAP f) l1 /\ happ (happ h4/list/MAP (happ (happ h4/combin/o g) h4/pair/SND)) (h4/list/ZIP (happ (happ h4/pair/_2C l1) l2)) = happ (happ h4/list/MAP g) l2
% Assm [h4s_arithmetics_DIVMODu_u_DEF]: !_4. (!f a m n q. happ (happ (happ (happ (happ _4 f) a) m) n) q = happ f (happ (happ h4/pair/_2C (h4/arithmetic/_2B a q)) (happ (happ h4/pair/_2C (h4/arithmetic/_2D m (h4/arithmetic/_2A n q))) n))) ==> (!_3. (!f a m n. ?v. (v <=> n = h4/num/0) /\ happ (happ (happ (happ _3 f) a) m) n = h4/bool/COND v (happ (happ h4/pair/_2C h4/num/0) h4/num/0) (h4/bool/COND (h4/prim__rec/_3C m n) (happ (happ h4/pair/_2C a) m) (h4/bool/LET (happ (happ (happ (happ _4 f) a) m) n) (h4/arithmetic/findq (happ (happ h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (happ (happ h4/pair/_2C m) n)))))) ==> (!_2. (!f a m. happ (happ (happ _2 f) a) m = happ (happ (happ _3 f) a) m) ==> (!_1. (!f a. happ (happ _1 f) a = h4/pair/UNCURRY (happ (happ _2 f) a)) ==> (!_0. (!f. happ _0 f = h4/pair/UNCURRY (happ _1 f)) ==> h4/arithmetic/DIVMOD = h4/relation/WFREC (h4/prim__rec/measure (happ (happ h4/combin/o h4/pair/FST) h4/pair/SND)) _0))))
% 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 (happ (happ h4/combin/o h4/combin/C) f)) x)
% Assm [h4s_pairs_FORALLu_u_UNCURRY]: !f. happ $forall (h4/pair/UNCURRY f) <=> happ $forall (happ (happ h4/combin/o $forall) f)
% Assm [h4s_stateu_u_transformers_BINDu_u_DEF]: !g f. h4/state__transformer/BIND g f = happ (happ h4/combin/o (h4/pair/UNCURRY f)) g
% Assm [h4s_stateu_u_transformers_mapMu_u_def]: !f. h4/state__transformer/mapM f = happ (happ h4/combin/o h4/state__transformer/sequence) (happ h4/list/MAP f)
% Assm [h4s_pairs_ou_u_UNCURRYu_u_R]: !g f. happ (happ h4/combin/o f) (h4/pair/UNCURRY g) = h4/pair/UNCURRY (happ (happ h4/combin/o (happ h4/combin/o f)) g)
% Assm [h4s_pairs_Su_u_UNCURRYu_u_R]: !g f. happ (happ h4/combin/S f) (h4/pair/UNCURRY g) = h4/pair/UNCURRY (happ (happ h4/combin/S (happ (happ h4/combin/o h4/combin/S) (happ (happ h4/combin/o (happ h4/combin/o f)) h4/pair/_2C))) g)
% Assm [h4s_lists_TLu_u_GENLIST]: !n f. h4/list/TL (h4/list/GENLIST f (happ h4/num/SUC n)) = h4/list/GENLIST (happ (happ h4/combin/o f) h4/num/SUC) n
% 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_sums_SUMu_u_MAPu_u_CASE]: !z g f. h4/sum/_2B_2B f g z = h4/sum/sum__CASE z (happ (happ h4/combin/o h4/sum/INL) f) (happ (happ h4/combin/o h4/sum/INR) g)
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c1]: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c0]: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_lists_splitAtPkiu_u_DEFu_c1]: !_1. (!k h p s. happ (happ (happ (happ _1 k) h) p) s = happ (happ k (happ (happ h4/list/CONS h) p)) s) ==> (!_0. (!k h p. happ (happ (happ _0 k) h) p = happ (happ (happ _1 k) h) p) ==> (!t k h P. h4/list/splitAtPki P k (happ (happ h4/list/CONS h) t) = h4/bool/COND (happ (happ P h4/num/0) h) (happ (happ k h4/list/NIL) (happ (happ h4/list/CONS h) t)) (h4/list/splitAtPki (happ (happ h4/combin/o P) h4/num/SUC) (happ (happ _0 k) h) t)))
% Assm [h4s_combins_GENu_u_LETu_u_RAND]: !v f P. happ P (h4/bool/LET f v) = h4/bool/LET (happ (happ h4/combin/o P) f) v
% Assm [h4s_lists_DROPu_u_splitAtPki]: !n l. h4/list/DROP n l = h4/list/splitAtPki (happ (happ h4/combin/o h4/combin/K) ($equals n)) (happ h4/combin/K h4/combin/I) l
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% 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_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_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 (happ h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% 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 (happ 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_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_pairs_SND0]: !y x. happ h4/pair/SND (happ (happ h4/pair/_2C x) y) = y
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (happ (happ h4/pair/_2C x) y) = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_lists_MAP0u_c1]: !t h f. happ (happ h4/list/MAP f) (happ (happ h4/list/CONS h) t) = happ (happ h4/list/CONS (happ f h)) (happ (happ h4/list/MAP f) t)
% Assm [h4s_lists_MAP0u_c0]: !f. happ (happ h4/list/MAP f) h4/list/NIL = h4/list/NIL
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_lists_MAPu_u_LISTu_u_BIND]: !l g f. happ (happ h4/list/MAP f) (h4/list/LIST__BIND l g) = h4/list/LIST__BIND l (happ (happ h4/combin/o (happ h4/list/MAP f)) g)
% Assm [h4s_lists_LISTu_u_APPLYu_u_DEF]: !xs fs. h4/list/LIST__APPLY fs xs = h4/list/LIST__BIND fs (happ (happ h4/combin/C h4/list/MAP) xs)
% Assm [h4s_lists_SINGLu_u_LISTu_u_APPLYu_u_L]: !x f. h4/list/LIST__BIND (happ (happ h4/list/CONS x) h4/list/NIL) f = happ f x
% Assm [h4s_lists_LISTu_u_BINDu_u_MAP]: !l g f. h4/list/LIST__BIND (happ (happ h4/list/MAP f) l) g = h4/list/LIST__BIND l (happ (happ h4/combin/o g) f)
% Assm [h4s_lists_LISTu_u_BINDu_u_APPEND]: !l2 l1 f. h4/list/LIST__BIND (h4/list/APPEND l1 l2) f = h4/list/APPEND (h4/list/LIST__BIND l1 f) (h4/list/LIST__BIND l2 f)
% Assm [h4s_lists_LISTu_u_BINDu_u_THMu_c0]: !f. h4/list/LIST__BIND h4/list/NIL f = h4/list/NIL
% Assm [h4s_lists_LISTu_u_BINDu_u_THMu_c1]: !t h f. h4/list/LIST__BIND (happ (happ h4/list/CONS h) t) f = h4/list/APPEND (happ f h) (h4/list/LIST__BIND t f)
% Assm [h4s_lists_APPENDu_u_11u_c1]: !l3 l2 l1. h4/list/APPEND l2 l1 = h4/list/APPEND l3 l1 <=> l2 = l3
% Assm [h4s_lists_MAPu_u_MAPu_u_o]: !l g f. happ (happ h4/list/MAP f) (happ (happ h4/list/MAP g) l) = happ (happ h4/list/MAP (happ (happ h4/combin/o f) g)) l
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% 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_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (happ h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_pairs_FORALLu_u_PROD]: !P. (!p. happ P p) <=> (!p__1 p__2. happ P (happ (happ h4/pair/_2C p__1) p__2))
% Assm [h4s_pairs_PAIR]: !x. happ (happ h4/pair/_2C (happ h4/pair/FST x)) (happ h4/pair/SND x) = x
% Assm [h4s_combins_Su_u_THM]: !x g f. happ (happ (happ h4/combin/S f) g) x = happ (happ f x) (happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_lists_ZIP0u_c0]: h4/list/ZIP (happ (happ h4/pair/_2C h4/list/NIL) h4/list/NIL) = h4/list/NIL
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (happ (happ h4/list/CONS h) t) = happ h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_lists_ZIP0u_c1]: !x2 x1 l2 l1. h4/list/ZIP (happ (happ h4/pair/_2C (happ (happ h4/list/CONS x1) l1)) (happ (happ h4/list/CONS x2) l2)) = happ (happ h4/list/CONS (happ (happ h4/pair/_2C x1) x2)) (h4/list/ZIP (happ (happ h4/pair/_2C l1) l2))
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. happ h4/num/SUC m = happ h4/num/SUC n <=> m = n
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(happ h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_sums_INRu_u_INLu_u_11u_c0]: !y x. happ h4/sum/INL x = happ h4/sum/INL y <=> x = y
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c1]: !g f b. h4/sum/_2B_2B f g (happ h4/sum/INR b) = happ h4/sum/INR (happ g b)
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c0]: !g f a. h4/sum/_2B_2B f g (happ h4/sum/INL a) = happ h4/sum/INL (happ f a)
% Assm [h4s_sums_INRu_u_INLu_u_11u_c1]: !y x. happ h4/sum/INR x = happ h4/sum/INR y <=> x = y
% Assm [h4s_sums_FORALLu_u_SUM]: !P. (!s. happ P s) <=> (!x. happ P (happ h4/sum/INL x)) /\ (!y. happ P (happ h4/sum/INR y))
% Assm [h4s_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (happ h4/sum/INL x) f f1 = happ f x
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (happ h4/sum/INR y) f f1 = happ f1 y
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. happ h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_lists_ELu_u_GENLIST]: !x n f. h4/prim__rec/_3C x n ==> h4/list/EL x (h4/list/GENLIST f n) = happ f x
% Assm [h4s_lists_LENGTHu_u_GENLIST]: !n f. h4/list/LENGTH (h4/list/GENLIST f n) = n
% Assm [h4s_lists_LISTu_u_EQ]: !l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ (!x. h4/prim__rec/_3C x (h4/list/LENGTH l1) ==> h4/list/EL x l1 = h4/list/EL x l2) ==> l1 = l2
% Assm [h4s_lists_LENGTHu_u_TL]: !l. h4/prim__rec/_3C h4/num/0 (h4/list/LENGTH l) ==> h4/list/LENGTH (h4/list/TL l) = h4/arithmetic/_2D (h4/list/LENGTH l) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_lists_EL0u_c1]: !n l. h4/list/EL (happ h4/num/SUC n) l = h4/list/EL n (h4/list/TL l)
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (happ h4/num/SUC m) n
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (happ h4/num/SUC n)
% Assm [h4s_arithmetics_SUCu_u_SUB1]: !m. h4/arithmetic/_2D (happ h4/num/SUC m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_lists_DROPu_u_LENGTHu_u_TOOu_u_LONG]: !n l. h4/arithmetic/_3C_3D (h4/list/LENGTH l) n ==> h4/list/DROP n l = h4/list/NIL
% Assm [h4s_lists_splitAtPkiu_u_EQN]: !_1. (!k l i. happ (happ (happ _1 k) l) i = happ (happ k (h4/list/TAKE i l)) (h4/list/DROP i l)) ==> (!_0. (!P l i. happ (happ (happ _0 P) l) i <=> h4/prim__rec/_3C i (h4/list/LENGTH l) /\ happ (happ P i) (h4/list/EL i l)) ==> (!l k P. h4/list/splitAtPki P k l = h4/option/option__CASE (h4/while/OLEAST (happ (happ _0 P) l)) (happ (happ k l) h4/list/NIL) (happ (happ _1 k) l)))
% Assm [h4s_whiles_OLEASTu_u_INTRO]: !Q P. ((!n. ~happ P n) ==> happ Q h4/option/NONE) /\ (!n. happ P n /\ (!m. h4/prim__rec/_3C m n ==> ~happ P m) ==> happ Q (h4/option/SOME n)) ==> happ Q (h4/while/OLEAST P)
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (happ h4/num/SUC n) m
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Goal: happ (happ h4/combin/o h4/state__transformer/JOIN) (h4/state__transformer/MMAP h4/state__transformer/JOIN) = happ (happ h4/combin/o h4/state__transformer/JOIN) h4/state__transformer/JOIN
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_Q1448462,TV_Q1448458]: ![V_f, V_g]: (![V_x]: s(TV_Q1448458,happ(s(t_fun(TV_Q1448462,TV_Q1448458),V_f),s(TV_Q1448462,V_x))) = s(TV_Q1448458,happ(s(t_fun(TV_Q1448462,TV_Q1448458),V_g),s(TV_Q1448462,V_x))) => s(t_fun(TV_Q1448462,TV_Q1448458),V_f) = s(t_fun(TV_Q1448462,TV_Q1448458),V_g))).
fof(ah4s_stateu_u_transformers_JOINu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_z]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join),s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),V_z))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),V_z),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_combins_i)))).
fof(ah4s_stateu_u_transformers_MMAPu_u_JOIN, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f]: s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27b,TV_u_27c),V_f))))),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join))) = s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),h4s_stateu_u_transformers_join))),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27b,TV_u_27c),V_f)))))))).
fof(ah4s_stateu_u_transformers_JOINu_u_MMAPu_u_UNIT, axiom, ![TV_u_27b,TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join))),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_unit))))) = s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_combins_i)).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_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),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_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),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_stateu_u_transformers_MMAPu_u_DEF, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_m, V_f]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),V_m))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),V_m),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),h4s_combins_o),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_unit))),s(t_fun(TV_u_27c,TV_u_27b),V_f)))))).
fof(ah4s_stateu_u_transformers_BINDu_u_LEFTu_u_UNIT, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_k]: s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_stateu_u_transformers_unit),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_k))) = s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_k),s(TV_u_27a,V_x)))).
fof(ah4s_stateu_u_transformers_JOINu_u_UNIT, axiom, ![TV_u_27b,TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join))),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),h4s_stateu_u_transformers_unit))) = s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_combins_i)).
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_EQu_u_EXT, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (![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))) => s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g))).
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_stateu_u_transformers_BINDu_u_ASSOC, axiom, ![TV_u_27b,TV_u_27c,TV_u_27d,TV_u_27a]: ![V_uu_0]: (![V_m, V_n, V_a]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_m))),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),V_n))),s(TV_u_27b,V_a))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_m),s(TV_u_27b,V_a))),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),V_n))) => ![V_n, V_m, V_k]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_k),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_m))),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),V_n))))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_k),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_m))),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),V_n))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
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_stateu_u_transformers_BINDu_u_RIGHTu_u_UNIT, axiom, ![TV_u_27b,TV_u_27a]: ![V_k]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_k),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_unit))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_k)).
fof(ah4s_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,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(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x))),s(t_fun(TV_u_27a,TV_u_27b),V_x0))),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_combins_Ku_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_Iu_u_DEF, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i) = s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27a),TV_u_27a)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),t_fun(TV_u_27a,TV_u_27a))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27a),TV_u_27a)),h4s_combins_k))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_combins_k)))).
fof(ah4s_stateu_u_transformers_MMAPu_u_COMP, axiom, ![TV_u_27d,TV_u_27a,TV_u_27b,TV_u_27c]: ![V_g, V_f]: s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27b,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),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_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27d))),h4s_combins_o),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g))))) = s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27d,TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27c,TV_u_27d),V_f))))),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27b,TV_u_27c),V_g)))))).
fof(ah4s_stateu_u_transformers_MMAPu_u_UNIT, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f]: s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27b,TV_u_27c),V_f))))),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_unit))) = s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))))),h4s_combins_o),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),h4s_stateu_u_transformers_unit))),s(t_fun(TV_u_27b,TV_u_27c),V_f)))).
fof(ah4s_stateu_u_transformers_MMAPu_u_ID, axiom, ![TV_u_27b,TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(TV_u_27b,TV_u_27b),h4s_combins_i))) = s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_combins_i)).
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),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b)),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_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_lists_LISTu_u_APPLYu_u_o, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_xs, V_gs, V_fs]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27a)),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a))),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))),t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a))))),happ(s(t_fun(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a))),t_fun(t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))),t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))))),h4s_lists_cons),s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a))),h4s_combins_o))),s(t_h4s_lists_list(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(TV_u_27b,TV_u_27a)))),h4s_lists_nil))),s(t_h4s_lists_list(t_fun(TV_u_27c,TV_u_27a)),V_fs))),s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27c)),V_gs))),s(t_h4s_lists_list(TV_u_27b),V_xs))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(TV_u_27c,TV_u_27a)),V_fs),s(t_h4s_lists_list(TV_u_27c),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27c)),V_gs),s(t_h4s_lists_list(TV_u_27b),V_xs)))))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_o, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27c)),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)))),h4s_options_some(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o))),s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27b)),V_g))),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27a,TV_u_27b)),V_g),s(t_h4s_options_option(TV_u_27a),V_x)))))).
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_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_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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_h4s_lists_list(TV_u_27a),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(TV_u_27d)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_h4s_lists_list(TV_u_27b),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_arithmetics_DIVMODu_u_DEF, axiom, ![V_uu_4]: (![V_f, V_a, V_m, V_n, V_q]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_q))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),h4s_pairs_u_2c),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_q))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),h4s_pairs_u_2c),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_q))))))),s(t_h4s_nums_num,V_n))))))) => ![V_uu_3]: (![V_f, V_a, V_m, V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),h4s_pairs_u_2c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,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_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),h4s_pairs_u_2c),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_findq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),h4s_pairs_u_2c),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),h4s_pairs_u_2c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))))))))))) => ![V_uu_2]: (![V_f, V_a, V_m]: s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))) = s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))) => ![V_uu_1]: (![V_f, V_a]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))))) => ![V_uu_0]: (![V_f]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_divmod) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)),h4s_primu_u_recs_measure(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num))),h4s_combins_o),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num),h4s_pairs_fst))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_snd))))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0))))))))).
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))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),t_fun(TV_u_27a,t_fun(TV_u_27d,t_fun(TV_u_27b,TV_u_27c)))),happ(s(t_fun(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))),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),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_FORALLu_u_UNCURRY, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),d_forall),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f))))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),d_forall),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool))),h4s_combins_o),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),d_forall))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f)))))).
fof(ah4s_stateu_u_transformers_BINDu_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_g),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_f))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),h4s_pairs_uncurry(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_f))))),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_g)))).
fof(ah4s_stateu_u_transformers_mapMu_u_def, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_f]: s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b))),h4s_stateu_u_transformers_mapm(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f))) = s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b))),t_fun(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b))))),h4s_combins_o),s(t_fun(t_h4s_lists_list(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b))),h4s_stateu_u_transformers_sequence))),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))))),h4s_lists_map),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f)))))).
fof(ah4s_pairs_ou_u_UNCURRYu_u_R, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b,TV_u_27d]: ![V_g, V_f]: s(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),TV_u_27d),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c))),h4s_combins_o),s(t_fun(TV_u_27d,TV_u_27c),V_f))),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)),V_g))))) = 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_27d)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)))),h4s_combins_o),s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),h4s_combins_o),s(t_fun(TV_u_27d,TV_u_27c),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),V_g)))))).
fof(ah4s_pairs_Su_u_UNCURRYu_u_R, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b,TV_u_27d]: ![V_g, V_f]: s(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),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(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c))),h4s_combins_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),V_f))),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)),V_g))))) = 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_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(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c)),t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,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(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),h4s_combins_s))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c)))),happ(s(t_fun(t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27d,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(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c)))),h4s_combins_o),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c))))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),V_g)))))).
fof(ah4s_lists_TLu_u_GENLIST, axiom, ![TV_u_27a]: ![V_n, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_fun(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,TV_u_27a))),h4s_combins_o),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc))),s(t_h4s_nums_num,V_n)))).
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_sums_SUMu_u_MAPu_u_CASE, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_z, V_g, V_f]: 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),V_z))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27c,TV_u_27d)))),h4s_combins_o),s(t_fun(TV_u_27c,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_sums_inl))),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27d,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27c,TV_u_27d)))),h4s_combins_o),s(t_fun(TV_u_27d,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_sums_inr))),s(t_fun(TV_u_27b,TV_u_27d),V_g)))))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_some(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(t_h4s_options_option(TV_u_27b),V_x)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_APPLYu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_none),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_lists_splitAtPkiu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_k, V_h, V_p, V_s]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_1),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_p))),s(t_h4s_lists_list(TV_u_27b),V_s))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),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_h))),s(t_h4s_lists_list(TV_u_27b),V_p))))),s(t_h4s_lists_list(TV_u_27b),V_s))) => ![V_uu_0]: (![V_k, V_h, V_p]: s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_0),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_p))) = s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_1),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_p))) => ![V_t, V_k, V_h, V_P]: s(TV_u_27a,h4s_lists_splitatpki(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),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_h))),s(t_h4s_lists_list(TV_u_27b),V_t))))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27b,V_h))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))),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_h))),s(t_h4s_lists_list(TV_u_27b),V_t))))),s(TV_u_27a,h4s_lists_splitatpki(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)))),h4s_combins_o),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc))),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_0),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t)))))))).
fof(ah4s_combins_GENu_u_LETu_u_RAND, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_v, V_f, V_P]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,V_v))))) = s(TV_u_27a,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a))),h4s_combins_o),s(t_fun(TV_u_27b,TV_u_27a),V_P))),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(TV_u_27c,V_v)))).
fof(ah4s_lists_DROPu_u_splitAtPki, axiom, ![TV_u_27a]: ![V_n, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_drop(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),h4s_lists_splitatpki(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)))),h4s_combins_o),s(t_fun(t_bool,t_fun(TV_u_27a,t_bool)),h4s_combins_k))),s(t_fun(t_h4s_nums_num,t_bool),d_equals(s(t_h4s_nums_num,V_n))))),s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)))),h4s_combins_k),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_combins_i))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
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_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_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_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,happ(s(t_fun(t_h4s_nums_num,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_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_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,happ(s(t_fun(t_h4s_nums_num,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_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_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_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_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),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_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),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_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_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),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),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),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),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),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_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_lists_MAPu_u_LISTu_u_BIND, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_l, V_g, V_f]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),h4s_lists_map),s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27c),V_l),s(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27b)),V_g))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27c),V_l),s(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27b)),t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),t_fun(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27b)),t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a)))),h4s_combins_o),s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),h4s_lists_map),s(t_fun(TV_u_27b,TV_u_27a),V_f))))),s(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27b)),V_g)))))).
fof(ah4s_lists_LISTu_u_APPLYu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_xs, V_fs]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_apply(s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27a)),V_fs),s(t_h4s_lists_list(TV_u_27b),V_xs))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(t_fun(TV_u_27b,TV_u_27a)),V_fs),s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_fun(TV_u_27b,TV_u_27a),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_fun(TV_u_27b,TV_u_27a),t_h4s_lists_list(TV_u_27a)))),h4s_combins_c),s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),h4s_lists_map))),s(t_h4s_lists_list(TV_u_27b),V_xs)))))).
fof(ah4s_lists_SINGLu_u_LISTu_u_APPLYu_u_L, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(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_x))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f),s(TV_u_27b,V_x)))).
fof(ah4s_lists_LISTu_u_BINDu_u_MAP, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_l, V_g, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27b))),h4s_lists_map),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27c),V_l))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_g))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27c),V_l),s(t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_lists_list(TV_u_27a)))),h4s_combins_o),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_f)))))).
fof(ah4s_lists_LISTu_u_BINDu_u_APPEND, axiom, ![TV_u_27b,TV_u_27a]: ![V_l2, V_l1, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),V_l1),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),V_l2),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f)))))).
fof(ah4s_lists_LISTu_u_BINDu_u_THMu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_LISTu_u_BINDu_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(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_h))),s(t_h4s_lists_list(TV_u_27b),V_t))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),V_t),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f)))))).
fof(ah4s_lists_APPENDu_u_11u_c1, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(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_append(s(t_h4s_lists_list(TV_u_27a),V_l3),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_l3))).
fof(ah4s_lists_MAPu_u_MAPu_u_o, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_l, V_g, V_f]: s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c))),h4s_lists_map),s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_lists_map),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27c))),h4s_lists_map),s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_o),s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g))))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_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_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_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_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_bools_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_pairs_FORALLu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))) <=> ![V_pu_u_1, V_pu_u_2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_pu_u_1))),s(TV_u_27b,V_pu_u_2)))))))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),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_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),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_combins_Su_u_THM, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,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(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = 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,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
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_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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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_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,happ(s(t_fun(t_h4s_nums_num,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_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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))) <=> 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,happ(s(t_fun(t_h4s_nums_num,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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => 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),happ(s(t_fun(TV_u_27a,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),happ(s(t_fun(TV_u_27a,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_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),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_b))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27d,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),happ(s(t_fun(TV_u_27a,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),happ(s(t_fun(TV_u_27c,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_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),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_y))).
fof(ah4s_sums_FORALLu_u_SUM, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_s]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_s)))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x)))))) & ![V_y]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))))))))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_lists_ELu_u_GENLIST, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n)))) => s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),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_x))))).
fof(ah4s_lists_LENGTHu_u_GENLIST, axiom, ![TV_u_27a]: ![V_n, V_f]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_lists_LISTu_u_EQ, axiom, ![TV_u_27a]: ![V_l2, V_l1]: ((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_27a),V_l2))) & ![V_x]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1)))))) => s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),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),V_l2))).
fof(ah4s_lists_LENGTHu_u_TL, axiom, ![TV_u_27a]: ![V_l]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_lists_EL0u_c1, axiom, ![TV_u_27a]: ![V_n, V_l]: s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: 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_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_SUCu_u_SUB1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_lists_DROPu_u_LENGTHu_u_TOOu_u_LONG, axiom, ![TV_u_27a]: ![V_n, V_l]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))),s(t_h4s_nums_num,V_n)))) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_drop(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),h4s_lists_nil))).
fof(ah4s_lists_splitAtPkiu_u_EQN, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_k, V_l, V_i]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_1),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(t_h4s_lists_list(TV_u_27b),V_l))),s(t_h4s_nums_num,V_i))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),s(t_h4s_lists_list(TV_u_27b),h4s_lists_take(s(t_h4s_nums_num,V_i),s(t_h4s_lists_list(TV_u_27b),V_l))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_drop(s(t_h4s_nums_num,V_i),s(t_h4s_lists_list(TV_u_27b),V_l))))) => ![V_uu_0]: (![V_P, V_l, V_i]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_nums_num,t_bool))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_lists_list(TV_u_27b),V_l))),s(t_h4s_nums_num,V_i)))) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_nums_num,V_i))),s(TV_u_27b,h4s_lists_el(s(t_h4s_nums_num,V_i),s(t_h4s_lists_list(TV_u_27b),V_l)))))))) => ![V_l, V_k, V_P]: s(TV_u_27a,h4s_lists_splitatpki(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),s(t_h4s_lists_list(TV_u_27b),V_l))) = s(TV_u_27a,h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_nums_num,t_bool))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_lists_list(TV_u_27b),V_l))))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),s(t_h4s_lists_list(TV_u_27b),V_l))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))),s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_1),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(t_h4s_lists_list(TV_u_27b),V_l)))))))).
fof(ah4s_whiles_OLEASTu_u_INTRO, axiom, ![V_Q, V_P]: (((![V_n]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))) & ![V_n]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_n)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ch4s_stateu_u_transformers_JOINu_u_MAPu_u_JOIN, conjecture, ![TV_u_27b,TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join))),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),h4s_stateu_u_transformers_mmap(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join))))) = s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),h4s_combins_o),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join))),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),h4s_stateu_u_transformers_join)))).
