%   ORIGINAL: h4/sptree/lookup__insert
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% 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/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/bool/COND__RATOR: !x g f b. h4/bool/COND b f g x = h4/bool/COND b (f x) (g x)
% Assm: h4/bool/COND__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Assm: h4/bool/COND__EXPAND: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% 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/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/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/NOT__IS__SOME__EQ__NONE: !x. ~h4/option/IS__SOME x <=> x = h4/option/NONE
% Assm: h4/option/IF__EQUALS__OPTION_c0: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c1: !X P. h4/bool/COND P h4/option/NONE X = h4/option/NONE <=> h4/option/IS__SOME X ==> P
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC 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/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/prim__rec/measure__def: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/arithmetic/SUB_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% 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/EVEN0_c1: !n. h4/arithmetic/EVEN (h4/num/SUC n) <=> ~h4/arithmetic/EVEN n
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/arithmetic/TWO: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% 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/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/SUB__0_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/MULT__CLAUSES_c3: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/arithmetic/MULT__COMM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/arithmetic/RIGHT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/arithmetic/LEFT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm: h4/arithmetic/MULT__ASSOC: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/EVEN__ADD: !n m. h4/arithmetic/EVEN (h4/arithmetic/_2B m n) <=> h4/arithmetic/EVEN m <=> h4/arithmetic/EVEN n
% Assm: h4/arithmetic/EVEN__MULT: !n m. h4/arithmetic/EVEN (h4/arithmetic/_2A m n) <=> h4/arithmetic/EVEN m \/ h4/arithmetic/EVEN n
% Assm: h4/arithmetic/EQ__LESS__EQ: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% 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/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% 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/MULT__LESS__EQ__SUC: !p n m. h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2A (h4/num/SUC p) m) (h4/arithmetic/_2A (h4/num/SUC p) n)
% Assm: h4/arithmetic/SUB__LEFT__ADD: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% Assm: h4/arithmetic/SUB__RIGHT__ADD: !p n m. h4/arithmetic/_2B (h4/arithmetic/_2D m n) p = h4/bool/COND (h4/arithmetic/_3C_3D m n) p (h4/arithmetic/_2D (h4/arithmetic/_2B m p) n)
% Assm: h4/arithmetic/SUB__RIGHT__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2D m n) p <=> h4/arithmetic/_3C_3D m (h4/arithmetic/_2B n p)
% Assm: h4/arithmetic/SUB__RIGHT__LESS: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2D m n) p <=> h4/prim__rec/_3C m (h4/arithmetic/_2B n p) /\ h4/prim__rec/_3C h4/num/0 p
% Assm: h4/arithmetic/SUB__LEFT__EQ: !p n m. m = h4/arithmetic/_2D n p <=> h4/arithmetic/_2B m p = n \/ h4/arithmetic/_3C_3D m h4/num/0 /\ h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/SUB__RIGHT__EQ: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm: h4/arithmetic/DIVISION: !n. h4/prim__rec/_3C h4/num/0 n ==> (!k. k = h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/DIV k n) n) (h4/arithmetic/MOD k n) /\ h4/prim__rec/_3C (h4/arithmetic/MOD k n) n)
% Assm: h4/arithmetic/DIV__UNIQUE: !q n k. (?r. k = h4/arithmetic/_2B (h4/arithmetic/_2A q n) r /\ h4/prim__rec/_3C r n) ==> h4/arithmetic/DIV k n = q
% Assm: h4/arithmetic/DIV__LT__X: !z y x. h4/prim__rec/_3C h4/num/0 z ==> (h4/prim__rec/_3C (h4/arithmetic/DIV y z) x <=> h4/prim__rec/_3C y (h4/arithmetic/_2A x z))
% Assm: h4/numeral/numeral__suc_c0: h4/num/SUC h4/arithmetic/ZERO = h4/arithmetic/BIT1 h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__distrib_c3: !n. h4/arithmetic/_2A h4/num/0 n = h4/num/0
% Assm: h4/numeral/numeral__distrib_c13: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c33: !n. h4/arithmetic/EVEN (h4/arithmetic/NUMERAL n) <=> h4/arithmetic/EVEN n
% Assm: h4/numeral/numeral__distrib_c35: h4/arithmetic/EVEN h4/num/0
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c3: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c4: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c5: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT2 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c6: !n. h4/num/SUC (h4/arithmetic/_2B h4/arithmetic/ZERO n) = h4/num/SUC n
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lt_c1: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lte_c2: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__evenodd_c1: !n. h4/arithmetic/EVEN (h4/arithmetic/BIT2 n)
% Assm: h4/numeral/numeral__evenodd_c2: !n. ~h4/arithmetic/EVEN (h4/arithmetic/BIT1 n)
% Assm: h4/sptree/spt__repfns_c0: !a. h4/sptree/_20_40ind__typesptree4 (h4/sptree/_20_40ind__typesptree5 a) = a
% Assm: h4/sptree/spt__repfns_c1: !r. (\a0_27. !_27spt_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27spt_27 a0 /\ _27spt_27 a1) \/ (?a0 a1 a2. a0_270 = (\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) a10 (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM)))) a0 a1 a2 /\ _27spt_27 a0 /\ _27spt_27 a2) ==> _27spt_27 a0_270) ==> _27spt_27 a0_27) r <=> h4/sptree/_20_40ind__typesptree5 (h4/sptree/_20_40ind__typesptree4 r) = r
% Assm: h4/sptree/hidden____20__40ind____typesptree0____def: h4/sptree/_20_40ind__typesptree0 = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/sptree/hidden____20__40ind____typesptree1____def: h4/sptree/_20_40ind__typesptree1 = (\a. h4/sptree/_20_40ind__typesptree4 ((\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/sptree/hidden____20__40ind____typesptree2____def: h4/sptree/_20_40ind__typesptree2 = (\a0 a1. h4/sptree/_20_40ind__typesptree4 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/sptree/_20_40ind__typesptree5 a0) (h4/sptree/_20_40ind__typesptree5 a1)))
% Assm: h4/sptree/hidden____20__40ind____typesptree3____def: h4/sptree/_20_40ind__typesptree3 = (\a0 a1 a2. h4/sptree/_20_40ind__typesptree4 ((\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) a10 (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM)))) (h4/sptree/_20_40ind__typesptree5 a0) a1 (h4/sptree/_20_40ind__typesptree5 a2)))
% Assm: h4/sptree/LN0: h4/sptree/LN = h4/sptree/_20_40ind__typesptree0
% Assm: h4/sptree/LS0: h4/sptree/LS = h4/sptree/_20_40ind__typesptree1
% Assm: h4/sptree/BN0: h4/sptree/BN = h4/sptree/_20_40ind__typesptree2
% Assm: h4/sptree/BS0: h4/sptree/BS = h4/sptree/_20_40ind__typesptree3
% Assm: h4/sptree/spt__case__def_c0: !v f2 f1 f. h4/sptree/spt__CASE h4/sptree/LN v f f1 f2 = v
% Assm: h4/sptree/spt__case__def_c1: !v f2 f1 f a. h4/sptree/spt__CASE (h4/sptree/LS a) v f f1 f2 = f a
% Assm: h4/sptree/spt__case__def_c2: !v f2 f1 f a1 a0. h4/sptree/spt__CASE (h4/sptree/BN a0 a1) v f f1 f2 = f1 a0 a1
% Assm: h4/sptree/spt__case__def_c3: !v f2 f1 f a2 a1 a0. h4/sptree/spt__CASE (h4/sptree/BS a0 a1 a2) v f f1 f2 = f2 a0 a1 a2
% Assm: h4/sptree/lookup__tupled__primitive__def: h4/sptree/lookup__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!t2 t1 k. ~(k = h4/num/0) ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/bool/COND (h4/arithmetic/EVEN k) t1 t2)) (h4/pair/_2C k (h4/sptree/BN t1 t2))) /\ (!a t2 t1 k. ~(k = h4/num/0) ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/bool/COND (h4/arithmetic/EVEN k) t1 t2)) (h4/pair/_2C k (h4/sptree/BS t1 a t2))))) (\lookup__tupled a_27_27. h4/pair/pair__CASE a_27_27 (\k v1. h4/sptree/spt__CASE v1 (h4/combin/I h4/option/NONE) (\a. h4/combin/I (h4/bool/COND (k = h4/num/0) (h4/option/SOME a) h4/option/NONE)) (\t1 t2. h4/combin/I (h4/bool/COND (k = h4/num/0) h4/option/NONE (lookup__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/bool/COND (h4/arithmetic/EVEN k) t1 t2))))) (\t1_27 a_27 t2_27. h4/combin/I (h4/bool/COND (k = h4/num/0) (h4/option/SOME a_27) (lookup__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/bool/COND (h4/arithmetic/EVEN k) t1_27 t2_27)))))))
% Assm: h4/sptree/lookup__curried__def: !x1 x. h4/sptree/lookup x x1 = h4/sptree/lookup__tupled (h4/pair/_2C x x1)
% Assm: h4/sptree/insert__tupled__primitive__def: h4/sptree/insert__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!a k. ~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN)) (h4/pair/_2C k (h4/pair/_2C a h4/sptree/LN))) /\ (!a k. ~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN)) (h4/pair/_2C k (h4/pair/_2C a h4/sptree/LN))) /\ (!a_27 a k. ~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN)) (h4/pair/_2C k (h4/pair/_2C a (h4/sptree/LS a_27)))) /\ (!a_27 a k. ~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN)) (h4/pair/_2C k (h4/pair/_2C a (h4/sptree/LS a_27)))) /\ (!t1 t2 a k. ~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t2)) (h4/pair/_2C k (h4/pair/_2C a (h4/sptree/BN t1 t2)))) /\ (!t2 t1 a k. ~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t1)) (h4/pair/_2C k (h4/pair/_2C a (h4/sptree/BN t1 t2)))) /\ (!t2 a_27 t1 a k. ~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t1)) (h4/pair/_2C k (h4/pair/_2C a (h4/sptree/BS t1 a_27 t2)))) /\ (!a_27 t1 t2 a k. ~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t2)) (h4/pair/_2C k (h4/pair/_2C a (h4/sptree/BS t1 a_27 t2)))))) (\insert__tupled a_27_27_27. h4/pair/pair__CASE a_27_27_27 (\k v1. h4/pair/pair__CASE v1 (\a v3. h4/sptree/spt__CASE v3 (h4/combin/I (h4/bool/COND (k = h4/num/0) (h4/sptree/LS a) (h4/bool/COND (h4/arithmetic/EVEN k) (h4/sptree/BN (insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))) h4/sptree/LN) (h4/sptree/BN h4/sptree/LN (insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))))))) (\a_27. h4/combin/I (h4/bool/COND (k = h4/num/0) (h4/sptree/LS a) (h4/bool/COND (h4/arithmetic/EVEN k) (h4/sptree/BS (insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))) a_27 h4/sptree/LN) (h4/sptree/BS h4/sptree/LN a_27 (insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))))))) (\t1 t2. h4/combin/I (h4/bool/COND (k = h4/num/0) (h4/sptree/BS t1 a t2) (h4/bool/COND (h4/arithmetic/EVEN k) (h4/sptree/BN (insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t1))) t2) (h4/sptree/BN t1 (insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t2))))))) (\t1_27 a_27_27 t2_27. h4/combin/I (h4/bool/COND (k = h4/num/0) (h4/sptree/BS t1_27 a t2_27) (h4/bool/COND (h4/arithmetic/EVEN k) (h4/sptree/BS (insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t1_27))) a_27_27 t2_27) (h4/sptree/BS t1_27 a_27_27 (insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t2_27))))))))))
% Assm: h4/sptree/insert__curried__def: !x2 x1 x. h4/sptree/insert x x1 x2 = h4/sptree/insert__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm: h4/sptree/insert__ind: !P. (!k a. (~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) a h4/sptree/LN) /\ (~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) a h4/sptree/LN) ==> P k a h4/sptree/LN) /\ (!k a a_27. (~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) a h4/sptree/LN) /\ (~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) a h4/sptree/LN) ==> P k a (h4/sptree/LS a_27)) /\ (!k a t1 t2. (~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) a t1) /\ (~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) a t2) ==> P k a (h4/sptree/BN t1 t2)) /\ (!k a t1 a_27 t2. (~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) a t1) /\ (~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) a t2) ==> P k a (h4/sptree/BS t1 a_27 t2)) ==> (!v v1 v2. P v v1 v2)
% Goal: !v t k2 k1. h4/sptree/lookup k1 (h4/sptree/insert k2 v t) = h4/bool/COND (k1 = k2) (h4/option/SOME v) (h4/sptree/lookup k1 t)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% 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_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_bools_CONDu_u_RATOR]: !x g f b. happ (h4/bool/COND b f g) x = h4/bool/COND b (happ f x) (happ g x)
% Assm [h4s_bools_CONDu_u_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% Assm [h4s_bools_CONDu_u_EXPAND]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% 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_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_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_options_NOTu_u_ISu_u_SOMEu_u_EQu_u_NONE]: !x. ~h4/option/IS__SOME x <=> x = h4/option/NONE
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c0]: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c1]: !X P. h4/bool/COND P h4/option/NONE X = h4/option/NONE <=> h4/option/IS__SOME X ==> P
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (happ (h4/relation/inv__image R) f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC 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_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~happ (happ h4/prim__rec/_3C n) h4/num/0
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/num/SUC n)
% Assm [h4s_primu_u_recs_measureu_u_def]: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (happ h4/prim__rec/measure m)
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_arithmetics_SUBu_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/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_EVEN0u_c1]: !n. h4/arithmetic/EVEN (h4/num/SUC n) <=> ~h4/arithmetic/EVEN n
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_arithmetics_TWO]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% 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_arithmetics_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_SUBu_u_0u_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% 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_MULTu_u_CLAUSESu_c3]: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_arithmetics_MULTu_u_COMM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm [h4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_arithmetics_LEFTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm [h4s_arithmetics_MULTu_u_ASSOC]: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_EVENu_u_ADD]: !n m. h4/arithmetic/EVEN (h4/arithmetic/_2B m n) <=> h4/arithmetic/EVEN m <=> h4/arithmetic/EVEN n
% Assm [h4s_arithmetics_EVENu_u_MULT]: !n m. h4/arithmetic/EVEN (h4/arithmetic/_2A m n) <=> h4/arithmetic/EVEN m \/ h4/arithmetic/EVEN n
% Assm [h4s_arithmetics_EQu_u_LESSu_u_EQ]: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% 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_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% 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_MULTu_u_LESSu_u_EQu_u_SUC]: !p n m. h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2A (h4/num/SUC p) m) (h4/arithmetic/_2A (h4/num/SUC p) n)
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_ADD]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_ADD]: !p n m. h4/arithmetic/_2B (h4/arithmetic/_2D m n) p = h4/bool/COND (h4/arithmetic/_3C_3D m n) p (h4/arithmetic/_2D (h4/arithmetic/_2B m p) n)
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2D m n) p <=> h4/arithmetic/_3C_3D m (h4/arithmetic/_2B n p)
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_LESS]: !p n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/_2D m n)) p <=> happ (happ h4/prim__rec/_3C m) (h4/arithmetic/_2B n p) /\ happ (happ h4/prim__rec/_3C h4/num/0) p
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_EQ]: !p n m. m = h4/arithmetic/_2D n p <=> h4/arithmetic/_2B m p = n \/ h4/arithmetic/_3C_3D m h4/num/0 /\ h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_EQ]: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm [h4s_arithmetics_DIVISION]: !n. happ (happ h4/prim__rec/_3C h4/num/0) n ==> (!k. k = h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/DIV k n) n) (h4/arithmetic/MOD k n) /\ happ (happ h4/prim__rec/_3C (h4/arithmetic/MOD k n)) n)
% Assm [h4s_arithmetics_DIVu_u_UNIQUE]: !q n k. (?r. k = h4/arithmetic/_2B (h4/arithmetic/_2A q n) r /\ happ (happ h4/prim__rec/_3C r) n) ==> h4/arithmetic/DIV k n = q
% Assm [h4s_arithmetics_DIVu_u_LTu_u_X]: !z y x. happ (happ h4/prim__rec/_3C h4/num/0) z ==> (happ (happ h4/prim__rec/_3C (h4/arithmetic/DIV y z)) x <=> happ (happ h4/prim__rec/_3C y) (h4/arithmetic/_2A x z))
% Assm [h4s_numerals_numeralu_u_sucu_c0]: h4/num/SUC h4/arithmetic/ZERO = h4/arithmetic/BIT1 h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_distribu_c3]: !n. h4/arithmetic/_2A h4/num/0 n = h4/num/0
% Assm [h4s_numerals_numeralu_u_distribu_c13]: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/arithmetic/NUMERAL n) <=> happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) n
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c33]: !n. h4/arithmetic/EVEN (h4/arithmetic/NUMERAL n) <=> h4/arithmetic/EVEN n
% Assm [h4s_numerals_numeralu_u_distribu_c35]: h4/arithmetic/EVEN h4/num/0
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c3]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c4]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c5]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT2 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c6]: !n. h4/num/SUC (h4/arithmetic/_2B h4/arithmetic/ZERO n) = h4/num/SUC n
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_ltu_c1]: !n. happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) (h4/arithmetic/BIT2 n) <=> 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_lteu_c2]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_evenoddu_c1]: !n. h4/arithmetic/EVEN (h4/arithmetic/BIT2 n)
% Assm [h4s_numerals_numeralu_u_evenoddu_c2]: !n. ~h4/arithmetic/EVEN (h4/arithmetic/BIT1 n)
% Assm [h4s_sptrees_sptu_u_repfnsu_c0]: !a. h4/sptree/_20_40ind__typesptree4 (h4/sptree/_20_40ind__typesptree5 a) = a
% Assm [h4s_sptrees_sptu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27spt_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a _0) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27spt_27 a0 /\ happ _27spt_27 a1) \/ (?a0 a1 a2. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) a1 (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a2 _0)) /\ happ _27spt_27 a0 /\ happ _27spt_27 a2) ==> happ _27spt_27 a0_270) ==> happ _27spt_27 r) <=> h4/sptree/_20_40ind__typesptree5 (h4/sptree/_20_40ind__typesptree4 r) = r)
% Assm [h4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/sptree/_20_40ind__typesptree0 = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/sptree/_20_40ind__typesptree1 x = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x _0))
% Assm [h4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/sptree/_20_40ind__typesptree2 x) x' = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS (h4/sptree/_20_40ind__typesptree5 x) (h4/ind__type/FCONS (h4/sptree/_20_40ind__typesptree5 x') _0))))
% Assm [h4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x x'. happ (happ (happ h4/sptree/_20_40ind__typesptree3 x) x) x' = h4/sptree/_20_40ind__typesptree4 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) x (h4/ind__type/FCONS (h4/sptree/_20_40ind__typesptree5 x) (h4/ind__type/FCONS (h4/sptree/_20_40ind__typesptree5 x') _0))))
% Assm [h4s_sptrees_LN0]: h4/sptree/LN = h4/sptree/_20_40ind__typesptree0
% Assm [h4s_sptrees_LS0]: h4/sptree/LS = h4/sptree/_20_40ind__typesptree1
% Assm [h4s_sptrees_BN0]: h4/sptree/BN = h4/sptree/_20_40ind__typesptree2
% Assm [h4s_sptrees_BS0]: h4/sptree/BS = h4/sptree/_20_40ind__typesptree3
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c0]: !v f2 f1 f. h4/sptree/spt__CASE h4/sptree/LN v f f1 f2 = v
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c1]: !v f2 f1 f a. h4/sptree/spt__CASE (happ h4/sptree/LS a) v f f1 f2 = happ f a
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c2]: !v f2 f1 f a1 a0. h4/sptree/spt__CASE (happ (happ h4/sptree/BN a0) a1) v f f1 f2 = happ (happ f1 a0) a1
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c3]: !v f2 f1 f a2 a1 a0. h4/sptree/spt__CASE (happ (happ (happ h4/sptree/BS a0) a1) a2) v f f1 f2 = happ (happ (happ f2 a0) a1) a2
% Assm [h4s_sptrees_lookupu_u_tupledu_u_primitiveu_u_def]: !_10. (!a_27 lookup__tupled k t1_27 t2_27. ?v. (v <=> k = h4/num/0) /\ happ (happ (happ (happ (happ _10 a_27) lookup__tupled) k) t1_27) t2_27 = h4/combin/I (h4/bool/COND v (h4/option/SOME a_27) (happ lookup__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/bool/COND (h4/arithmetic/EVEN k) t1_27 t2_27))))) ==> (!_9. (!lookup__tupled k t1_27 a_27. happ (happ (happ (happ _9 lookup__tupled) k) t1_27) a_27 = happ (happ (happ (happ _10 a_27) lookup__tupled) k) t1_27) ==> (!_8. (!lookup__tupled k t1_27. happ (happ (happ _8 lookup__tupled) k) t1_27 = happ (happ (happ _9 lookup__tupled) k) t1_27) ==> (!_7. (!lookup__tupled k t1 t2. ?v. (v <=> k = h4/num/0) /\ happ (happ (happ (happ _7 lookup__tupled) k) t1) t2 = h4/combin/I (h4/bool/COND v h4/option/NONE (happ lookup__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/bool/COND (h4/arithmetic/EVEN k) t1 t2))))) ==> (!_6. (!lookup__tupled k t1. happ (happ (happ _6 lookup__tupled) k) t1 = happ (happ (happ _7 lookup__tupled) k) t1) ==> (!_5. (!k a. ?v. (v <=> k = h4/num/0) /\ happ (happ _5 k) a = h4/combin/I (h4/bool/COND v (h4/option/SOME a) h4/option/NONE)) ==> (!_4. (!lookup__tupled k v1. happ (happ (happ _4 lookup__tupled) k) v1 = h4/sptree/spt__CASE v1 (h4/combin/I h4/option/NONE) (happ _5 k) (happ (happ _6 lookup__tupled) k) (happ (happ _8 lookup__tupled) k)) ==> (!_3. (!lookup__tupled k. happ (happ _3 lookup__tupled) k = happ (happ _4 lookup__tupled) k) ==> (!_2. (!lookup__tupled a_27_27. happ (happ _2 lookup__tupled) a_27_27 = h4/pair/pair__CASE a_27_27 (happ _3 lookup__tupled)) ==> (!_1. (!lookup__tupled. happ _1 lookup__tupled = happ _2 lookup__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!t2 t1 k. ~(k = h4/num/0) ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/bool/COND (h4/arithmetic/EVEN k) t1 t2))) (h4/pair/_2C k (happ (happ h4/sptree/BN t1) t2))) /\ (!a t2 t1 k. ~(k = h4/num/0) ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/bool/COND (h4/arithmetic/EVEN k) t1 t2))) (h4/pair/_2C k (happ (happ (happ h4/sptree/BS t1) a) t2)))) ==> h4/sptree/lookup__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))
% Assm [h4s_sptrees_lookupu_u_curriedu_u_def]: !x1 x. h4/sptree/lookup x x1 = happ h4/sptree/lookup__tupled (h4/pair/_2C x x1)
% Assm [h4s_sptrees_insertu_u_tupledu_u_primitiveu_u_def]: !_12. (!t1_27 a_27_27 insert__tupled k a t2_27. ?v. (v <=> k = h4/num/0) /\ happ (happ (happ (happ (happ (happ _12 t1_27) a_27_27) insert__tupled) k) a) t2_27 = h4/combin/I (h4/bool/COND v (happ (happ (happ h4/sptree/BS t1_27) a) t2_27) (h4/bool/COND (h4/arithmetic/EVEN k) (happ (happ (happ h4/sptree/BS (happ insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t1_27)))) a_27_27) t2_27) (happ (happ (happ h4/sptree/BS t1_27) a_27_27) (happ insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t2_27))))))) ==> (!_11. (!t1_27 insert__tupled k a a_27_27. happ (happ (happ (happ (happ _11 t1_27) insert__tupled) k) a) a_27_27 = happ (happ (happ (happ (happ _12 t1_27) a_27_27) insert__tupled) k) a) ==> (!_10. (!insert__tupled k a t1_27. happ (happ (happ (happ _10 insert__tupled) k) a) t1_27 = happ (happ (happ (happ _11 t1_27) insert__tupled) k) a) ==> (!_9. (!t1 insert__tupled k a t2. ?v. (v <=> k = h4/num/0) /\ happ (happ (happ (happ (happ _9 t1) insert__tupled) k) a) t2 = h4/combin/I (h4/bool/COND v (happ (happ (happ h4/sptree/BS t1) a) t2) (h4/bool/COND (h4/arithmetic/EVEN k) (happ (happ h4/sptree/BN (happ insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t1)))) t2) (happ (happ h4/sptree/BN t1) (happ insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t2))))))) ==> (!_8. (!insert__tupled k a t1. happ (happ (happ (happ _8 insert__tupled) k) a) t1 = happ (happ (happ (happ _9 t1) insert__tupled) k) a) ==> (!_7. (!insert__tupled k a a_27. ?v. (v <=> k = h4/num/0) /\ happ (happ (happ (happ _7 insert__tupled) k) a) a_27 = h4/combin/I (h4/bool/COND v (happ h4/sptree/LS a) (h4/bool/COND (h4/arithmetic/EVEN k) (happ (happ (happ h4/sptree/BS (happ insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN)))) a_27) h4/sptree/LN) (happ (happ (happ h4/sptree/BS h4/sptree/LN) a_27) (happ insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))))))) ==> (!_6. (!insert__tupled k a v3. ?v. (v <=> k = h4/num/0) /\ happ (happ (happ (happ _6 insert__tupled) k) a) v3 = h4/sptree/spt__CASE v3 (h4/combin/I (h4/bool/COND v (happ h4/sptree/LS a) (h4/bool/COND (h4/arithmetic/EVEN k) (happ (happ h4/sptree/BN (happ insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN)))) h4/sptree/LN) (happ (happ h4/sptree/BN h4/sptree/LN) (happ insert__tupled (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))))))) (happ (happ (happ _7 insert__tupled) k) a) (happ (happ (happ _8 insert__tupled) k) a) (happ (happ (happ _10 insert__tupled) k) a)) ==> (!_5. (!insert__tupled k a. happ (happ (happ _5 insert__tupled) k) a = happ (happ (happ _6 insert__tupled) k) a) ==> (!_4. (!insert__tupled k v1. happ (happ (happ _4 insert__tupled) k) v1 = h4/pair/pair__CASE v1 (happ (happ _5 insert__tupled) k)) ==> (!_3. (!insert__tupled k. happ (happ _3 insert__tupled) k = happ (happ _4 insert__tupled) k) ==> (!_2. (!insert__tupled a_27_27_27. happ (happ _2 insert__tupled) a_27_27_27 = h4/pair/pair__CASE a_27_27_27 (happ _3 insert__tupled)) ==> (!_1. (!insert__tupled. happ _1 insert__tupled = happ _2 insert__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!a k. ~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))) (h4/pair/_2C k (h4/pair/_2C a h4/sptree/LN))) /\ (!a k. ~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))) (h4/pair/_2C k (h4/pair/_2C a h4/sptree/LN))) /\ (!a_27 a k. ~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))) (h4/pair/_2C k (h4/pair/_2C a (happ h4/sptree/LS a_27)))) /\ (!a_27 a k. ~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a h4/sptree/LN))) (h4/pair/_2C k (h4/pair/_2C a (happ h4/sptree/LS a_27)))) /\ (!t1 t2 a k. ~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t2))) (h4/pair/_2C k (h4/pair/_2C a (happ (happ h4/sptree/BN t1) t2)))) /\ (!t2 t1 a k. ~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t1))) (h4/pair/_2C k (h4/pair/_2C a (happ (happ h4/sptree/BN t1) t2)))) /\ (!t2 a_27 t1 a k. ~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t1))) (h4/pair/_2C k (h4/pair/_2C a (happ (happ (happ h4/sptree/BS t1) a_27) t2)))) /\ (!a_27 t1 t2 a k. ~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> happ (happ R (h4/pair/_2C (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/pair/_2C a t2))) (h4/pair/_2C k (h4/pair/_2C a (happ (happ (happ h4/sptree/BS t1) a_27) t2))))) ==> h4/sptree/insert__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))
% Assm [h4s_sptrees_insertu_u_curriedu_u_def]: !x2 x1 x. h4/sptree/insert x x1 x2 = happ h4/sptree/insert__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm [h4s_sptrees_insertu_u_ind]: !P. (!k a. (~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> happ (happ (happ P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) a) h4/sptree/LN) /\ (~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> happ (happ (happ P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) a) h4/sptree/LN) ==> happ (happ (happ P k) a) h4/sptree/LN) /\ (!k a a_27. (~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> happ (happ (happ P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) a) h4/sptree/LN) /\ (~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> happ (happ (happ P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) a) h4/sptree/LN) ==> happ (happ (happ P k) a) (happ h4/sptree/LS a_27)) /\ (!k a t1 t2. (~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> happ (happ (happ P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) a) t1) /\ (~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> happ (happ (happ P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) a) t2) ==> happ (happ (happ P k) a) (happ (happ h4/sptree/BN t1) t2)) /\ (!k a t1 a_27 t2. (~(k = h4/num/0) /\ h4/arithmetic/EVEN k ==> happ (happ (happ P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) a) t1) /\ (~(k = h4/num/0) /\ ~h4/arithmetic/EVEN k ==> happ (happ (happ P (h4/arithmetic/DIV (h4/arithmetic/_2D k (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) a) t2) ==> happ (happ (happ P k) a) (happ (happ (happ h4/sptree/BS t1) a_27) t2)) ==> (!v v1 v2. happ (happ (happ P v) v1) v2)
% Goal: !v t k2 k1. ?v. (v <=> k1 = k2) /\ h4/sptree/lookup k1 (h4/sptree/insert k2 v t) = h4/bool/COND v (h4/option/SOME v) (h4/sptree/lookup k1 t)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q81593,TV_Q81589]: ![V_f, V_g]: (![V_x]: s(TV_Q81589,happ(s(t_fun(TV_Q81593,TV_Q81589),V_f),s(TV_Q81593,V_x))) = s(TV_Q81589,happ(s(t_fun(TV_Q81593,TV_Q81589),V_g),s(TV_Q81593,V_x))) => s(t_fun(TV_Q81593,TV_Q81589),V_f) = s(t_fun(TV_Q81593,TV_Q81589),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t0))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
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,t0)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t0))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t0)))).
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,t0)))).
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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t0) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t0),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_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
fof(ah4s_bools_CONDu_u_RATOR, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_CONDu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))))).
fof(ah4s_bools_CONDu_u_EXPAND, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((~ (p(s(t_bool,V_b))) | p(s(t_bool,V_t1))) & (p(s(t_bool,V_b)) | p(s(t_bool,V_t2)))))).
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_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_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
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_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t0)).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,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_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_NOTu_u_ISu_u_SOMEu_u_EQu_u_NONE, axiom, ![TV_u_27a]: ![V_x]: (~ (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x))))) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c0, axiom, ![TV_u_27a]: ![V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> ~ (p(s(t_bool,V_P))))).
fof(ah4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c1, axiom, ![TV_u_27a]: ![V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),V_X))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_X)))) => p(s(t_bool,V_P))))).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_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_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_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_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_primu_u_recs_measureu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure) = s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c)))).
fof(ah4s_primu_u_recs_WFu_u_measure, axiom, ![TV_u_27a]: ![V_m]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure),s(t_fun(TV_u_27a,t_h4s_nums_num),V_m))))))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_arithmetics_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_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_EVEN0u_c1, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) <=> ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
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_arithmetics_TWO, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_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_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))).
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_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_SUBu_u_0u_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_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_MULTu_u_CLAUSESu_c3, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_LEFTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MULTu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_EVENu_u_ADD, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_even(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_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_m))) = s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_EVENu_u_MULT, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))) <=> (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_m)))) | p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_EQu_u_LESSu_u_EQ, axiom, ![V_n, V_m]: (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_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_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_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_NOTu_u_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (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,h4s_nums_suc(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,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))))))).
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_MULTu_u_LESSu_u_EQu_u_SUC, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_ADD, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_ADD, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_arithmetics_u_2d(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_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_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_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_LESS, axiom, ![V_p, 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,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_p)))) <=> (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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))) & 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_p))))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_EQ, axiom, ![V_p, V_n, V_m]: (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_p))) <=> (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_h4s_nums_num,V_n) | (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0)))) & 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_SUBu_u_RIGHTu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_p) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) | (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_DIVISION, 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,V_n)))) => ![V_k]: (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_DIVu_u_UNIQUE, axiom, ![V_q, V_n, V_k]: (?[V_r]: (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_r))) & 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_r))),s(t_h4s_nums_num,V_n))))) => s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_q))).
fof(ah4s_arithmetics_DIVu_u_LTu_u_X, axiom, ![V_z, V_y, V_x]: (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_z)))) => 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_arithmetics_div(s(t_h4s_nums_num,V_y),s(t_h4s_nums_num,V_z))))),s(t_h4s_nums_num,V_x))) = 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_y))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_z))))))).
fof(ah4s_numerals_numeralu_u_sucu_c0, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_distribu_c3, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_numerals_numeralu_u_distribu_c13, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (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_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![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,h4s_nums_0))),s(t_h4s_nums_num,h4s_arithmetics_numeral(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,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_distribu_c33, axiom, ![V_n]: s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c35, axiom, p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c4, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c5, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c6, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_equ_c1, 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_zero) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_ltu_c1, axiom, ![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,h4s_arithmetics_zero))),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))) = s(t_bool,t0)).
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_lteu_c2, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_evenoddu_c1, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_numerals_numeralu_u_evenoddu_c2, axiom, ![V_n]: ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_sptrees_sptu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),V_a))))) = s(t_h4s_sptrees_spt(TV_u_27a),V_a)).
fof(ah4s_sptrees_sptu_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_27sptu_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_a]: 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_a),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_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) | ?[V_a0, V_a1, V_a2]: (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_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(TV_u_27a,V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a2),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_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a2))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27sptu_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_27sptu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(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_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree0u_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_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree0) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(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_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree1u_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]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_u_20u_40indu_u_typesptree1),s(TV_u_27a,V_x))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(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)),V_uu_0))))))).
fof(ah4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree2u_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_xi_]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_u_20u_40indu_u_typesptree2),s(t_h4s_sptrees_spt(TV_u_27a),V_x))),s(t_h4s_sptrees_spt(TV_u_27a),V_xi_))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(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_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))))).
fof(ah4s_sptrees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typesptree3u_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, V_xi_]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_u_20u_40indu_u_typesptree3),s(t_h4s_sptrees_spt(TV_u_27a),V_x))),s(TV_u_27a,V_x0))),s(t_h4s_sptrees_spt(TV_u_27a),V_xi_))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree4(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(TV_u_27a,V_x0),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_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(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_sptrees_u_20u_40indu_u_typesptree5(s(t_h4s_sptrees_spt(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))))).
fof(ah4s_sptrees_LN0, axiom, ![TV_u_27a]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_u_20u_40indu_u_typesptree0)).
fof(ah4s_sptrees_LS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls) = s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_u_20u_40indu_u_typesptree1)).
fof(ah4s_sptrees_BN0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_u_20u_40indu_u_typesptree2)).
fof(ah4s_sptrees_BS0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_u_20u_40indu_u_typesptree3)).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f2, V_f1, V_f]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,V_v)).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls),s(TV_u_27a,V_a))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a)))).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),V_a0))),s(t_h4s_sptrees_spt(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_h4s_sptrees_spt(TV_u_27a),V_a0))),s(t_h4s_sptrees_spt(TV_u_27a),V_a1)))).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_a0))),s(TV_u_27a,V_a1))),s(t_h4s_sptrees_spt(TV_u_27a),V_a2))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2),s(t_h4s_sptrees_spt(TV_u_27a),V_a0))),s(TV_u_27a,V_a1))),s(t_h4s_sptrees_spt(TV_u_27a),V_a2)))).
fof(ah4s_sptrees_lookupu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_10]: (![V_au_27, V_lookupu_u_tupled, V_k, V_t1u_27, V_t2u_27]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_10),s(TV_u_27a,V_au_27))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(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_some(s(TV_u_27a,V_au_27))),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27)))))))))))) => ![V_uu_9]: (![V_lookupu_u_tupled, V_k, V_t1u_27, V_au_27]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_au_27))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_10),s(TV_u_27a,V_au_27))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))) => ![V_uu_8]: (![V_lookupu_u_tupled, V_k, V_t1u_27]: s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))) = s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))) => ![V_uu_7]: (![V_lookupu_u_tupled, V_k, V_t1, V_t2]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(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_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))))))))))) => ![V_uu_6]: (![V_lookupu_u_tupled, V_k, V_t1]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))) => ![V_uu_5]: (![V_k, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),V_uu_5),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(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_some(s(TV_u_27a,V_a))),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))) => ![V_uu_4]: (![V_lookupu_u_tupled, V_k, V_v1]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))) = s(t_h4s_options_option(TV_u_27a),h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),V_v1),s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),V_uu_5),s(t_h4s_nums_num,V_k))),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))))) => ![V_uu_3]: (![V_lookupu_u_tupled, V_k]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_nums_num,V_k))) => ![V_uu_2]: (![V_lookupu_u_tupled, V_au_27u_27]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),V_au_27u_27))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),V_au_27u_27),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))))) => ![V_uu_1]: (![V_lookupu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),V_lookupu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool)),V_R)))) & (![V_t2, V_t1, V_k]: (~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))))) & ![V_a, V_t2, V_t1, V_k]: (~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))))))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),h4s_sptrees_lookupu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a))),V_uu_1))))))))))))))).
fof(ah4s_sptrees_lookupu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x1, V_x]: s(t_h4s_options_option(TV_u_27a),h4s_sptrees_lookup(s(t_h4s_nums_num,V_x),s(t_h4s_sptrees_spt(TV_u_27a),V_x1))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_options_option(TV_u_27a)),h4s_sptrees_lookupu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_sptrees_spt(TV_u_27a),V_x1)))))).
fof(ah4s_sptrees_insertu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_12]: (![V_t1u_27, V_au_27u_27, V_insertu_u_tupled, V_k, V_a, V_t2u_27]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))))),V_uu_12),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_au_27u_27))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))))))))),s(TV_u_27a,V_au_27u_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_au_27u_27))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t2u_27)))))))))))))))) => ![V_uu_11]: (![V_t1u_27, V_insertu_u_tupled, V_k, V_a, V_au_27u_27]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))))),V_uu_11),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(TV_u_27a,V_au_27u_27))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))))),V_uu_12),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(TV_u_27a,V_au_27u_27))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))) => ![V_uu_10]: (![V_insertu_u_tupled, V_k, V_a, V_t1u_27]: s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))) = s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))))),V_uu_11),s(t_h4s_sptrees_spt(TV_u_27a),V_t1u_27))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))) => ![V_uu_9]: (![V_t1, V_insertu_u_tupled, V_k, V_a, V_t2]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),V_uu_9),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))))))))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))))))))))))))) => ![V_uu_8]: (![V_insertu_u_tupled, V_k, V_a, V_t1]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),V_uu_8),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),V_uu_9),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))) => ![V_uu_7]: (![V_insertu_u_tupled, V_k, V_a, V_au_27]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(TV_u_27a,V_au_27))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))))),s(TV_u_27a,V_au_27))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))),s(TV_u_27a,V_au_27))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln)))))))))))))))) => ![V_uu_6]: (![V_insertu_u_tupled, V_k, V_a, V_v3]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_v3))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),V_v3),s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))))))))))),s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),V_uu_8),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a)))))) => ![V_uu_5]: (![V_insertu_u_tupled, V_k, V_a]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))) => ![V_uu_4]: (![V_insertu_u_tupled, V_k, V_v1]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),V_v1))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),V_v1),s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))))) => ![V_uu_3]: (![V_insertu_u_tupled, V_k]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_nums_num,V_k))) => ![V_uu_2]: (![V_insertu_u_tupled, V_au_27u_27u_27]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),V_au_27u_27u_27))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),V_au_27u_27u_27),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))))) => ![V_uu_1]: (![V_insertu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),V_insertu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R)))) & (![V_a, V_k]: ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))))) & (![V_a, V_k]: ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))))) & (![V_au_27, V_a, V_k]: ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls),s(TV_u_27a,V_au_27))))))))))) & (![V_au_27, V_a, V_k]: ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls),s(TV_u_27a,V_au_27))))))))))) & (![V_t1, V_t2, V_a, V_k]: ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))))))) & (![V_t2, V_t1, V_a, V_k]: ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))))))) & (![V_t2, V_au_27, V_t1, V_a, V_k]: ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(TV_u_27a,V_au_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))))))) & ![V_au_27, V_t1, V_t2, V_a, V_k]: ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(TV_u_27a,V_au_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))))))))))))))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_insertu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a))),V_uu_1))))))))))))))))).
fof(ah4s_sptrees_insertu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x2, V_x1, V_x]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_insert(s(t_h4s_nums_num,V_x),s(TV_u_27a,V_x1),s(t_h4s_sptrees_spt(TV_u_27a),V_x2))) = s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_insertu_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_x1),s(t_h4s_sptrees_spt(TV_u_27a),V_x2)))))))).
fof(ah4s_sptrees_insertu_u_ind, axiom, ![TV_u_27a]: ![V_P]: ((![V_k, V_a]: ((((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))) & ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))) & (![V_k, V_a, V_au_27]: ((((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))))) & ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_ls),s(TV_u_27a,V_au_27))))))) & (![V_k, V_a, V_t1, V_t2]: ((((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))))) & ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),h4s_sptrees_bn),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))))))) & ![V_k, V_a, V_t1, V_au_27, V_t2]: ((((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))))) & ((~ (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_k)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_k),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,V_k))),s(TV_u_27a,V_a))),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),h4s_sptrees_bs),s(t_h4s_sptrees_spt(TV_u_27a),V_t1))),s(TV_u_27a,V_au_27))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))))))))) => ![V_v, V_v1, V_v2]: p(s(t_bool,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_bool))),V_P),s(t_h4s_nums_num,V_v))),s(TV_u_27a,V_v1))),s(t_h4s_sptrees_spt(TV_u_27a),V_v2)))))).
fof(ch4s_sptrees_lookupu_u_insert, conjecture, ![TV_u_27a]: ![V_v, V_t, V_k2, V_k1]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_h4s_nums_num,V_k1) = s(t_h4s_nums_num,V_k2)) & s(t_h4s_options_option(TV_u_27a),h4s_sptrees_lookup(s(t_h4s_nums_num,V_k1),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_insert(s(t_h4s_nums_num,V_k2),s(TV_u_27a,V_v),s(t_h4s_sptrees_spt(TV_u_27a),V_t))))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v0),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_v))),s(t_h4s_options_option(TV_u_27a),h4s_sptrees_lookup(s(t_h4s_nums_num,V_k1),s(t_h4s_sptrees_spt(TV_u_27a),V_t))))))).
