%   ORIGINAL: h4/path/nth__label__def__compute_c2
% 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/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/arithmetic/SUC__ELIM__NUMERALS: !g f. (!n. g (h4/num/SUC n) = f n (h4/num/SUC n)) <=> (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/path/nth__label__def_c1: !p n. h4/path/nth__label (h4/num/SUC n) p = h4/path/nth__label n (h4/path/tail p)
% Assm: h4/path/el__def_c1: !p n. h4/path/el (h4/num/SUC n) p = h4/path/el n (h4/path/tail p)
% Assm: h4/path/nth__label__def_c0: !p. h4/path/nth__label h4/num/0 p = h4/path/first__label p
% Assm: h4/path/nth__label__def__compute_c1: !p n. h4/path/nth__label (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) p = h4/path/nth__label (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/path/tail p)
% Assm: h4/path/el__def_c0: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm: h4/path/el__def__compute_c2: !p n. h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) p = h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/path/tail p)
% Assm: h4/path/el__def__compute_c1: !p n. h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) p = h4/path/el (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/path/tail p)
% Assm: h4/path/nth__label__def__compute_c0: !p. h4/path/nth__label h4/num/0 p = h4/path/first__label p
% Assm: h4/path/tail__def: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/list/EL__simp_c1: !n l. h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) l = h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/list/TL l)
% Assm: h4/list/GENLIST__AUX0_c1: !n l f. h4/list/GENLIST__AUX f (h4/num/SUC n) l = h4/list/GENLIST__AUX f n (h4/list/CONS (f n) l)
% Assm: h4/list/GENLIST__AUX0_c0: !l f. h4/list/GENLIST__AUX f h4/num/0 l = l
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/list/LUPDATE__compute_c3: !x n l e. h4/list/LUPDATE e (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) (h4/list/CONS x l) = h4/list/CONS x (h4/list/LUPDATE e (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) l)
% Assm: h4/list/TAKE__compute_c4: !t n h. h4/list/TAKE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) (h4/list/CONS h t) = h4/list/CONS h (h4/list/TAKE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) t)
% Assm: h4/rich__list/REPLICATE__compute_c2: !x n. h4/rich__list/REPLICATE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) x = h4/list/CONS x (h4/rich__list/REPLICATE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) x)
% Assm: h4/arithmetic/EXP__SUB__NUMERAL: !x n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x))) n = h4/arithmetic/EXP n (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ h4/arithmetic/DIV (h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 x))) n = h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x))
% Assm: h4/rich__list/SEG__compute_c5: !x m l k. h4/rich__list/SEG (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 m)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 k)) (h4/list/CONS x l) = h4/rich__list/SEG (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 m)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 k)) l
% Assm: h4/numpair/tri__def__compute_c2: !n. h4/numpair/tri (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) (h4/numpair/tri (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))
% Assm: h4/list/EL__simp__restricted_c1: !n ls l. h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) (h4/list/CONS l ls) = h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) ls
% Assm: h4/rich__list/SEG__compute_c6: !x m l k. h4/rich__list/SEG (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 m)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 k)) (h4/list/CONS x l) = h4/rich__list/SEG (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 m)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 k)) l
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/list/GENLIST__AUX__compute_c2: !n l f. h4/list/GENLIST__AUX f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) l = h4/list/GENLIST__AUX f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/list/CONS (f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) l)
% Assm: h4/list/GENLIST__AUX__compute0_c2: !n l f. h4/list/GENLIST__AUX f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) l = h4/list/GENLIST__AUX f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/list/CONS (f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) l)
% Assm: h4/set__relation/nth__min__def__compute_c2: !s r_27 r n. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = h4/bool/LET (\min. h4/bool/COND (min = h4/option/NONE) h4/option/NONE (h4/set__relation/nth__min r_27 (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))) (h4/set__relation/get__min r_27 (h4/pair/_2C s r))
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/numpair/tri__def_c1: !n. h4/numpair/tri (h4/num/SUC n) = h4/arithmetic/_2B (h4/num/SUC n) (h4/numpair/tri n)
% Assm: h4/numpair/tri__def_c0: h4/numpair/tri h4/num/0 = h4/num/0
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/path/pcons__11: !y x s r q p. h4/path/pcons x r p = h4/path/pcons y s q <=> x = y /\ r = s /\ p = q
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/bool/SELECT__REFL__2: !x. h4/min/_40 (\y. x = y) = x
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/rich__list/SEG0_c0: !l k. h4/rich__list/SEG h4/num/0 k l = h4/list/NIL
% Assm: h4/rich__list/SEG0_c2: !x m l k. h4/rich__list/SEG (h4/num/SUC m) (h4/num/SUC k) (h4/list/CONS x l) = h4/rich__list/SEG (h4/num/SUC m) k l
% Assm: h4/rich__list/SEG0_c1: !x m l. h4/rich__list/SEG (h4/num/SUC m) h4/num/0 (h4/list/CONS x l) = h4/list/CONS x (h4/rich__list/SEG m h4/num/0 l)
% Assm: h4/rich__list/REPLICATE0_c1: !x n. h4/rich__list/REPLICATE (h4/num/SUC n) x = h4/list/CONS x (h4/rich__list/REPLICATE n x)
% Assm: h4/rich__list/REPLICATE0_c0: !x. h4/rich__list/REPLICATE h4/num/0 x = h4/list/NIL
% Assm: h4/list/LUPDATE__def_c2: !x n l e. h4/list/LUPDATE e (h4/num/SUC n) (h4/list/CONS x l) = h4/list/CONS x (h4/list/LUPDATE e n l)
% Assm: h4/list/LUPDATE__def_c0: !n e. h4/list/LUPDATE e n h4/list/NIL = h4/list/NIL
% Assm: h4/list/LUPDATE__def_c1: !x l e. h4/list/LUPDATE e h4/num/0 (h4/list/CONS x l) = h4/list/CONS e l
% Assm: h4/list/EL__simp_c0: !n l. h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) l = h4/list/EL (h4/prim__rec/PRE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (h4/list/TL l)
% Assm: h4/list/TL0: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/BIT20: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm: h4/arithmetic/BIT10: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm: h4/arithmetic/NUMERAL__DEF: !x. h4/arithmetic/NUMERAL x = x
% 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/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/EXP__1_c1: !n. h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = n
% Assm: h4/arithmetic/SUB__MONO__EQ: !n m. h4/arithmetic/_2D (h4/num/SUC n) (h4/num/SUC m) = h4/arithmetic/_2D n m
% Assm: h4/arithmetic/LESS__EQ__MONO: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/SUB__0_c1: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm: h4/arithmetic/EXP__SUB: !q p n. h4/prim__rec/_3C h4/num/0 n /\ h4/arithmetic/_3C_3D q p ==> h4/arithmetic/EXP n (h4/arithmetic/_2D p q) = h4/arithmetic/DIV (h4/arithmetic/EXP n p) (h4/arithmetic/EXP n q)
% Assm: h4/arithmetic/ALT__ZERO: h4/arithmetic/ZERO = h4/num/0
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/list/TAKE__0: !l. h4/list/TAKE h4/num/0 l = h4/list/NIL
% Assm: h4/list/TAKE__def_c1: !xs x n. h4/list/TAKE n (h4/list/CONS x xs) = h4/bool/COND (n = h4/num/0) h4/list/NIL (h4/list/CONS x (h4/list/TAKE (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) xs))
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% 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/TAKE__def_c0: !n. h4/list/TAKE n h4/list/NIL = h4/list/NIL
% 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/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/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% 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/arithmetic/SUB_c1: !n m. h4/arithmetic/_2D (h4/num/SUC m) n = h4/bool/COND (h4/prim__rec/_3C m n) h4/num/0 (h4/num/SUC (h4/arithmetic/_2D m n))
% Assm: h4/arithmetic/SUB_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = 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/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/set__relation/nth__min__curried__def: !x2 x1 x. h4/set__relation/nth__min x x1 x2 = h4/set__relation/nth__min__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/set__relation/nth__min__tupled__primitive__def: h4/set__relation/nth__min__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!n r s r_27 min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> R (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)) (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C s r) (h4/num/SUC n)))))) (\nth__min__tupled a. h4/pair/pair__CASE a (\r_27 v1. h4/pair/pair__CASE v1 (\v2 v3. h4/pair/pair__CASE v2 (\s r. h4/arithmetic/num__CASE v3 (h4/combin/I (h4/set__relation/get__min r_27 (h4/pair/_2C s r))) (\n. h4/combin/I (h4/bool/LET (\min. h4/bool/COND (min = h4/option/NONE) h4/option/NONE (nth__min__tupled (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)))) (h4/set__relation/get__min r_27 (h4/pair/_2C s r))))))))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% 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/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/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/num__case__def_c1: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = f n
% Assm: h4/arithmetic/num__case__def_c0: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/bool/LET__CONG: !g f N M. M = N /\ (!x. x = N ==> f x = g x) ==> h4/bool/LET f M = h4/bool/LET g N
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Goal: !p n. h4/path/nth__label (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) p = h4/path/nth__label (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/path/tail p)
%   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_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS]: !g f. (!n. happ g (h4/num/SUC n) = happ (happ f n) (h4/num/SUC n)) <=> (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = happ (happ f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = happ (happ f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_paths_nthu_u_labelu_u_defu_c1]: !p n. h4/path/nth__label (h4/num/SUC n) p = h4/path/nth__label n (h4/path/tail p)
% Assm [h4s_paths_elu_u_defu_c1]: !p n. h4/path/el (h4/num/SUC n) p = h4/path/el n (h4/path/tail p)
% Assm [h4s_paths_nthu_u_labelu_u_defu_c0]: !p. h4/path/nth__label h4/num/0 p = h4/path/first__label p
% Assm [h4s_paths_nthu_u_labelu_u_defu_u_computeu_c1]: !p n. h4/path/nth__label (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) p = h4/path/nth__label (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/path/tail p)
% Assm [h4s_paths_elu_u_defu_c0]: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm [h4s_paths_elu_u_defu_u_computeu_c2]: !p n. h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) p = h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/path/tail p)
% Assm [h4s_paths_elu_u_defu_u_computeu_c1]: !p n. h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) p = h4/path/el (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/path/tail p)
% Assm [h4s_paths_nthu_u_labelu_u_defu_u_computeu_c0]: !p. h4/path/nth__label h4/num/0 p = h4/path/first__label p
% Assm [h4s_paths_tailu_u_def]: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_lists_ELu_u_simpu_c1]: !n l. h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) l = h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/list/TL l)
% Assm [h4s_lists_GENLISTu_u_AUX0u_c1]: !n l f. h4/list/GENLIST__AUX f (h4/num/SUC n) l = h4/list/GENLIST__AUX f n (happ (happ h4/list/CONS (happ f n)) l)
% Assm [h4s_lists_GENLISTu_u_AUX0u_c0]: !l f. h4/list/GENLIST__AUX f h4/num/0 l = l
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_lists_LUPDATEu_u_computeu_c3]: !x n l e. h4/list/LUPDATE e (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) (happ (happ h4/list/CONS x) l) = happ (happ h4/list/CONS x) (h4/list/LUPDATE e (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) l)
% Assm [h4s_lists_TAKEu_u_computeu_c4]: !t n h. h4/list/TAKE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) (happ (happ h4/list/CONS h) t) = happ (happ h4/list/CONS h) (h4/list/TAKE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) t)
% Assm [h4s_richu_u_lists_REPLICATEu_u_computeu_c2]: !x n. h4/rich__list/REPLICATE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) x = happ (happ h4/list/CONS x) (h4/rich__list/REPLICATE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) x)
% Assm [h4s_arithmetics_EXPu_u_SUBu_u_NUMERAL]: !x n. happ (happ h4/prim__rec/_3C h4/num/0) n ==> h4/arithmetic/DIV (h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x))) n = h4/arithmetic/EXP n (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ h4/arithmetic/DIV (h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 x))) n = h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x))
% Assm [h4s_richu_u_lists_SEGu_u_computeu_c5]: !x m l k. h4/rich__list/SEG (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 m)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 k)) (happ (happ h4/list/CONS x) l) = h4/rich__list/SEG (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 m)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 k)) l
% Assm [h4s_numpairs_triu_u_defu_u_computeu_c2]: !n. h4/numpair/tri (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) (h4/numpair/tri (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))
% Assm [h4s_lists_ELu_u_simpu_u_restrictedu_c1]: !n ls l. h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) (happ (happ h4/list/CONS l) ls) = h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) ls
% Assm [h4s_richu_u_lists_SEGu_u_computeu_c6]: !x m l k. h4/rich__list/SEG (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 m)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 k)) (happ (happ h4/list/CONS x) l) = h4/rich__list/SEG (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 m)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 k)) l
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_lists_GENLISTu_u_AUXu_u_computeu_c2]: !n l f. h4/list/GENLIST__AUX f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) l = h4/list/GENLIST__AUX f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (happ (happ h4/list/CONS (happ f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))) l)
% Assm [h4s_lists_GENLISTu_u_AUXu_u_compute0u_c2]: !n l f. h4/list/GENLIST__AUX f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) l = h4/list/GENLIST__AUX f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (happ (happ h4/list/CONS (happ f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))) l)
% Assm [h4s_setu_u_relations_nthu_u_minu_u_defu_u_computeu_c2]: !_0. (!r_27 s r n min. ?v. (v <=> min = h4/option/NONE) /\ happ (happ (happ (happ (happ _0 r_27) s) r) n) min = h4/bool/COND v h4/option/NONE (h4/set__relation/nth__min r_27 (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))) ==> (!s r_27 r n. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = h4/bool/LET (happ (happ (happ (happ _0 r_27) s) r) n) (h4/set__relation/get__min r_27 (h4/pair/_2C s r)))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_numpairs_triu_u_defu_c1]: !n. h4/numpair/tri (h4/num/SUC n) = h4/arithmetic/_2B (h4/num/SUC n) (h4/numpair/tri n)
% Assm [h4s_numpairs_triu_u_defu_c0]: h4/numpair/tri h4/num/0 = h4/num/0
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_paths_pconsu_u_11]: !y x s r q p. h4/path/pcons x r p = h4/path/pcons y s q <=> x = y /\ r = s /\ p = q
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_bools_SELECTu_u_REFLu_u_2]: !_0. (!x y. happ (happ _0 x) y <=> x = y) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_richu_u_lists_SEG0u_c0]: !l k. h4/rich__list/SEG h4/num/0 k l = h4/list/NIL
% Assm [h4s_richu_u_lists_SEG0u_c2]: !x m l k. h4/rich__list/SEG (h4/num/SUC m) (h4/num/SUC k) (happ (happ h4/list/CONS x) l) = h4/rich__list/SEG (h4/num/SUC m) k l
% Assm [h4s_richu_u_lists_SEG0u_c1]: !x m l. h4/rich__list/SEG (h4/num/SUC m) h4/num/0 (happ (happ h4/list/CONS x) l) = happ (happ h4/list/CONS x) (h4/rich__list/SEG m h4/num/0 l)
% Assm [h4s_richu_u_lists_REPLICATE0u_c1]: !x n. h4/rich__list/REPLICATE (h4/num/SUC n) x = happ (happ h4/list/CONS x) (h4/rich__list/REPLICATE n x)
% Assm [h4s_richu_u_lists_REPLICATE0u_c0]: !x. h4/rich__list/REPLICATE h4/num/0 x = h4/list/NIL
% Assm [h4s_lists_LUPDATEu_u_defu_c2]: !x n l e. h4/list/LUPDATE e (h4/num/SUC n) (happ (happ h4/list/CONS x) l) = happ (happ h4/list/CONS x) (h4/list/LUPDATE e n l)
% Assm [h4s_lists_LUPDATEu_u_defu_c0]: !n e. h4/list/LUPDATE e n h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_LUPDATEu_u_defu_c1]: !x l e. h4/list/LUPDATE e h4/num/0 (happ (happ h4/list/CONS x) l) = happ (happ h4/list/CONS e) l
% Assm [h4s_lists_ELu_u_simpu_c0]: !n l. h4/list/EL (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) l = h4/list/EL (h4/prim__rec/PRE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (h4/list/TL l)
% Assm [h4s_lists_TL0]: !t h. h4/list/TL (happ (happ h4/list/CONS h) t) = t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_BIT20]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm [h4s_arithmetics_BIT10]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm [h4s_arithmetics_NUMERALu_u_DEF]: !x. h4/arithmetic/NUMERAL x = x
% Assm [h4s_lists_EL0u_c1]: !n l. h4/list/EL (h4/num/SUC n) l = h4/list/EL n (h4/list/TL l)
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_EXPu_u_1u_c1]: !n. h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = n
% Assm [h4s_arithmetics_SUBu_u_MONOu_u_EQ]: !n m. h4/arithmetic/_2D (h4/num/SUC n) (h4/num/SUC m) = h4/arithmetic/_2D n m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_MONO]: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_SUBu_u_0u_c1]: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm [h4s_arithmetics_EXPu_u_SUB]: !q p n. happ (happ h4/prim__rec/_3C h4/num/0) n /\ h4/arithmetic/_3C_3D q p ==> h4/arithmetic/EXP n (h4/arithmetic/_2D p q) = h4/arithmetic/DIV (h4/arithmetic/EXP n p) (h4/arithmetic/EXP n q)
% Assm [h4s_arithmetics_ALTu_u_ZERO]: h4/arithmetic/ZERO = h4/num/0
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_lists_TAKEu_u_0]: !l. h4/list/TAKE h4/num/0 l = h4/list/NIL
% Assm [h4s_lists_TAKEu_u_defu_c1]: !xs x n. ?v. (v <=> n = h4/num/0) /\ h4/list/TAKE n (happ (happ h4/list/CONS x) xs) = h4/bool/COND v h4/list/NIL (happ (happ h4/list/CONS x) (h4/list/TAKE (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) xs))
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% 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_TAKEu_u_defu_c0]: !n. h4/list/TAKE n h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_listu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 r) <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r)
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/num/SUC n)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C (h4/num/SUC m)) (h4/num/SUC n) <=> happ (happ h4/prim__rec/_3C m) n
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_arithmetics_SUBu_c1]: !n m. h4/arithmetic/_2D (h4/num/SUC m) n = h4/bool/COND (happ (happ h4/prim__rec/_3C m) n) h4/num/0 (h4/num/SUC (h4/arithmetic/_2D m n))
% Assm [h4s_arithmetics_SUBu_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~happ (happ h4/prim__rec/_3C n) h4/num/0
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = 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_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_setu_u_relations_nthu_u_minu_u_curriedu_u_def]: !x2 x1 x. h4/set__relation/nth__min x x1 x2 = happ h4/set__relation/nth__min__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_setu_u_relations_nthu_u_minu_u_tupledu_u_primitiveu_u_def]: !_10. (!nth__min__tupled r_27 s r n min. ?v. (v <=> min = h4/option/NONE) /\ happ (happ (happ (happ (happ (happ _10 nth__min__tupled) r_27) s) r) n) min = h4/bool/COND v h4/option/NONE (happ nth__min__tupled (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)))) ==> (!_9. (!nth__min__tupled r_27 s r n. happ (happ (happ (happ (happ _9 nth__min__tupled) r_27) s) r) n = h4/combin/I (h4/bool/LET (happ (happ (happ (happ (happ _10 nth__min__tupled) r_27) s) r) n) (h4/set__relation/get__min r_27 (h4/pair/_2C s r)))) ==> (!_8. (!v3 nth__min__tupled r_27 s r. happ (happ (happ (happ (happ _8 v3) nth__min__tupled) r_27) s) r = h4/arithmetic/num__CASE v3 (h4/combin/I (h4/set__relation/get__min r_27 (h4/pair/_2C s r))) (happ (happ (happ (happ _9 nth__min__tupled) r_27) s) r)) ==> (!_7. (!v3 nth__min__tupled r_27 s. happ (happ (happ (happ _7 v3) nth__min__tupled) r_27) s = happ (happ (happ (happ _8 v3) nth__min__tupled) r_27) s) ==> (!_6. (!v2 nth__min__tupled r_27 v3. happ (happ (happ (happ _6 v2) nth__min__tupled) r_27) v3 = h4/pair/pair__CASE v2 (happ (happ (happ _7 v3) nth__min__tupled) r_27)) ==> (!_5. (!nth__min__tupled r_27 v2. happ (happ (happ _5 nth__min__tupled) r_27) v2 = happ (happ (happ _6 v2) nth__min__tupled) r_27) ==> (!_4. (!nth__min__tupled r_27 v1. happ (happ (happ _4 nth__min__tupled) r_27) v1 = h4/pair/pair__CASE v1 (happ (happ _5 nth__min__tupled) r_27)) ==> (!_3. (!nth__min__tupled r_27. happ (happ _3 nth__min__tupled) r_27 = happ (happ _4 nth__min__tupled) r_27) ==> (!_2. (!nth__min__tupled a. happ (happ _2 nth__min__tupled) a = h4/pair/pair__CASE a (happ _3 nth__min__tupled)) ==> (!_1. (!nth__min__tupled. happ _1 nth__min__tupled = happ _2 nth__min__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!n r s r_27 min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> happ (happ R (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n))) (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C s r) (h4/num/SUC n))))) ==> h4/set__relation/nth__min__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% 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_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_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_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c1]: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = happ f n
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c0]: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_bools_LETu_u_CONG]: !g f N M. M = N /\ (!x. x = N ==> happ f x = happ g x) ==> h4/bool/LET f M = h4/bool/LET g N
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Goal: !p n. h4/path/nth__label (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) p = h4/path/nth__label (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/path/tail p)
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_Q1228146,TV_Q1228142]: ![V_f, V_g]: (![V_x]: s(TV_Q1228142,happ(s(t_fun(TV_Q1228146,TV_Q1228142),V_f),s(TV_Q1228146,V_x))) = s(TV_Q1228142,happ(s(t_fun(TV_Q1228146,TV_Q1228142),V_g),s(TV_Q1228146,V_x))) => s(t_fun(TV_Q1228146,TV_Q1228142),V_f) = s(t_fun(TV_Q1228146,TV_Q1228142),V_g))).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,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_fn),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS, axiom, ![TV_u_27a]: ![V_g, V_f]: (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) <=> (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))))))))).
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_paths_nthu_u_labelu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p))) = s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,V_n),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p)))))).
fof(ah4s_paths_elu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_n),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_nthu_u_labelu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_p]: s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p))) = s(TV_u_27a,h4s_paths_firstu_u_label(s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p)))).
fof(ah4s_paths_nthu_u_labelu_u_defu_u_computeu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p))) = s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),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_paths_path(TV_u_27b,TV_u_27a),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p)))))).
fof(ah4s_paths_elu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))).
fof(ah4s_paths_elu_u_defu_u_computeu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_elu_u_defu_u_computeu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),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_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_nthu_u_labelu_u_defu_u_computeu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_p]: s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p))) = s(TV_u_27a,h4s_paths_firstu_u_label(s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p)))).
fof(ah4s_paths_tailu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_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_lists_ELu_u_simpu_c1, axiom, ![TV_u_27a]: ![V_n, V_l]: s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(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,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(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_lists_GENLISTu_u_AUX0u_c1, axiom, ![TV_u_27a]: ![V_n, V_l, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlistu_u_aux(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlistu_u_aux(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),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,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_GENLISTu_u_AUX0u_c0, axiom, ![TV_u_27a]: ![V_l, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlistu_u_aux(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
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_LUPDATEu_u_computeu_c3, axiom, ![TV_u_27a]: ![V_x, V_n, V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_TAKEu_u_computeu_c4, axiom, ![TV_u_27c]: ![V_t, V_n, V_h]: s(t_h4s_lists_list(TV_u_27c),h4s_lists_take(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))),s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(TV_u_27c,t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c))),h4s_lists_cons),s(TV_u_27c,V_h))),s(t_h4s_lists_list(TV_u_27c),V_t))))) = s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(TV_u_27c,t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c))),h4s_lists_cons),s(TV_u_27c,V_h))),s(t_h4s_lists_list(TV_u_27c),h4s_lists_take(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_lists_list(TV_u_27c),V_t)))))).
fof(ah4s_richu_u_lists_REPLICATEu_u_computeu_c2, axiom, ![TV_u_27a]: ![V_x, V_n]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))),s(TV_u_27a,V_x))) = 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_x))),s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(TV_u_27a,V_x)))))).
fof(ah4s_arithmetics_EXPu_u_SUBu_u_NUMERAL, axiom, ![V_x, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_n)))) => (s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) & s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_x))))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x)))))))))).
fof(ah4s_richu_u_lists_SEGu_u_computeu_c5, axiom, ![TV_u_27a]: ![V_x, V_m, V_l, V_k]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_k))))),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(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,V_k))))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_numpairs_triu_u_defu_u_computeu_c2, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(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_bit2(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_lists_ELu_u_simpu_u_restrictedu_c1, axiom, ![TV_u_27a]: ![V_n, V_ls, V_l]: s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))),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_l))),s(t_h4s_lists_list(TV_u_27a),V_ls))))) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_lists_list(TV_u_27a),V_ls)))).
fof(ah4s_richu_u_lists_SEGu_u_computeu_c6, axiom, ![TV_u_27a]: ![V_x, V_m, V_l, V_k]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_k))))),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(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,V_k))))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
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_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_lists_GENLISTu_u_AUXu_u_computeu_c2, axiom, ![TV_u_27a]: ![V_n, V_l, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlistu_u_aux(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(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_genlistu_u_aux(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),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,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_GENLISTu_u_AUXu_u_compute0u_c2, axiom, ![TV_u_27a]: ![V_n, V_l, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlistu_u_aux(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(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_genlistu_u_aux(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),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,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_defu_u_computeu_c2, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_ru_27, V_s, V_r, V_n, V_min]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),V_min))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n)))))))))) => ![V_s, V_ru_27, V_r, V_n]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_let(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))))).
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_numpairs_triu_u_defu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_numpairs_triu_u_defu_c0, axiom, s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_CONJu_u_ASSOC, 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_paths_pconsu_u_11, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_s, V_r, V_q, V_p]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_y),s(TV_u_27b,V_s),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) & (s(TV_u_27b,V_r) = s(TV_u_27b,V_s) & s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
fof(ah4s_bools_SELECTu_u_REFLu_u_2, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_richu_u_lists_SEG0u_c0, axiom, ![TV_u_27a]: ![V_l, V_k]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_k),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_richu_u_lists_SEG0u_c2, axiom, ![TV_u_27a]: ![V_x, V_m, V_l, V_k]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_k))),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_k),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_richu_u_lists_SEG0u_c1, axiom, ![TV_u_27a]: ![V_x, V_m, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_0),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_richu_u_lists_REPLICATE0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = 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_x))),s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x)))))).
fof(ah4s_richu_u_lists_REPLICATE0u_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_LUPDATEu_u_defu_c2, axiom, ![TV_u_27a]: ![V_x, V_n, V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_LUPDATEu_u_defu_c0, axiom, ![TV_u_27a]: ![V_n, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_LUPDATEu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,h4s_nums_0),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_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_lists_ELu_u_simpu_c0, axiom, ![TV_u_27a]: ![V_n, V_l]: s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(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,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(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_lists_TL0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),V_t)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, 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,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_BIT20, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit2(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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_arithmetics_BIT10, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit1(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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_NUMERALu_u_DEF, axiom, ![V_x]: s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
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,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_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = 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_EXPu_u_1u_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),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_SUBu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_MONO, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_SUBu_u_0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_EXPu_u_SUB, axiom, ![V_q, V_p, V_n]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_p))))) => s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_q))))) = s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_q))))))).
fof(ah4s_arithmetics_ALTu_u_ZERO, axiom, s(t_h4s_nums_num,h4s_arithmetics_zero) = s(t_h4s_nums_num,h4s_nums_0)).
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_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_bools_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_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_TAKEu_u_0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_take(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_TAKEu_u_defu_c1, axiom, ![TV_u_27a]: ![V_xs, V_x, 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_lists_list(TV_u_27a),h4s_lists_take(s(t_h4s_nums_num,V_n),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_x))),s(t_h4s_lists_list(TV_u_27a),V_xs))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_take(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),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_lists_list(TV_u_27a),V_xs))))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_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_TAKEu_u_defu_c0, axiom, ![TV_u_27a]: ![V_n]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_take(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_listu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27listu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
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_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_arithmetics_SUBu_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_arithmetics_SUBu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_ONE, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_primu_u_recs_NOTu_u_LESSu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![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_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![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)))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x2, V_x1, V_x]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_x),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_x1),s(t_h4s_nums_num,V_x2))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),h4s_setu_u_relations_nthu_u_minu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_x),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_x1),s(t_h4s_nums_num,V_x2)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_10]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r, V_n, V_min]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),V_min))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n)))))))))) => ![V_uu_9]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r, V_n]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_bools_let(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))))) => ![V_uu_8]: (![V_v3, V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))) = s(t_h4s_options_option(TV_u_27a),h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,V_v3),s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) => ![V_uu_7]: (![V_v3, V_nthu_u_minu_u_tupled, V_ru_27, V_s]: s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_7),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))) => ![V_uu_6]: (![V_v2, V_nthu_u_minu_u_tupled, V_ru_27, V_v3]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_nums_num,V_v3))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2),s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_7),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))))) => ![V_uu_5]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_v2]: s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))) = s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) => ![V_uu_4]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_v1]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),V_v1))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),V_v1),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))))) => ![V_uu_3]: (![V_nthu_u_minu_u_tupled, V_ru_27]: s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) = s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) => ![V_uu_2]: (![V_nthu_u_minu_u_tupled, V_a]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),V_a))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),V_a),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))))) => ![V_uu_1]: (![V_nthu_u_minu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R)))) & ![V_n, V_r, V_s, V_ru_27, V_min]: ((s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) & ~ (s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))))))),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))))))))) => s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),h4s_setu_u_relations_nthu_u_minu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_1))))))))))))))).
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_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_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_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_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_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_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_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_numu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = 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_arithmetics_numu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,V_v)).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_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_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(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_bools_LETu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_N, V_M]: ((s(TV_u_27a,V_M) = s(TV_u_27a,V_N) & ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_N) => 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(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_M))) = s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_N))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_IMPu_u_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(ch4s_paths_nthu_u_labelu_u_defu_u_computeu_c2, conjecture, ![TV_u_27b,TV_u_27a]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p))) = s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p)))))).
