Fixpoint arities (e:BC) : Arities :=
  match e with
  | zero => ok_arities 0 0
  | proj n s i => 
    if leb (S i) (n + s) then ok_arities n s
      else error_proj n s i
  | succ _ => ok_arities 0 1
  | pred => ok_arities 0 1
  | cond => ok_arities 0 4
  | rec g h0 h1 =>
      match arities g, arities h0, arities h1 
        with | ok_arities gn gs, ok_arities h0n h0s, ok_arities h1n h1s =>
          if beq_nat (S gn) h0n && beq_nat h0n h1n &&
            beq_nat (S gs) h0s && beq_nat h0s h1s 
          then ok_arities h0n gs 
            else error_rec (ok_arities gn gs) (ok_arities h0n h0s) (ok_arities h1n h1s)
        | e1, e2, e3 => error_rec e1 e2 e3
      end
    | comp n s h nl sl =>
      match arities h, map arities nl, map arities sl with
      | ok_arities hn hs, anl, asl =>
        if beq_nat hn (length nl) && beq_nat hs (length sl) && 
          forallb (fun ne => aeq (arities ne) (ok_arities n 0)) nl &&
          forallb (fun se => aeq (arities se) (ok_arities n s)) sl
          then ok_arities n s
          else error_comp (ok_arities hn hs) anl asl 
      | e , anl, asl => error_comp e anl asl
      end
  end.Definition arities2 e :=
  match arities e with
    | ok_arities n s => (n, s)
    | _ => (0, 0)
  end.

Lemma BC_ind_inf : forall (P : nat -> nat -> BC -> Prop), P 0 0 zero -> (forall n s i, i < n + s -> P n s (proj n s i)) -> (forall b, P 0 1 (succ b)) -> P 0 1 pred -> P 0 4 cond -> (forall n s g h0 h1, arities g = ok_arities n s -> arities h0 = ok_arities (S n) (S s) -> arities h1 = ok_arities (S n) (S s) -> P n s g -> P (S n) (S s) h0 -> P (S n) (S s) h1 -> P (S n) s (rec g h0 h1)) -> (forall n s h rl tl, arities h = ok_arities (length rl) (length tl) -> (forall e, In e rl -> arities e = ok_arities n 0) -> (forall e, In e tl -> arities e = ok_arities n s) -> P (length rl) (length tl) h -> (forall r, In r rl -> P n 0 r) -> (forall r, In r tl -> P n s r) -> P n s (comp n s h rl tl)) -> forall e n s , arities e = ok_arities n s -> P n s e.