Fixpoint arity (e : Cobham) : Arity :=
  match e with
    | Zero => ok_arity 0
    | Proj n j => if leb (S j) n 
      then ok_arity n else error_Proj n j 
    | Succ _ => ok_arity 1
    | Smash => ok_arity 2
    | Rec g h0 h1 j => 
      match arity g, arity h0, arity h1, arity j with
        | ok_arity gn, ok_arity h0n, ok_arity h1n, ok_arity jn =>
          if beq_nat (S (S gn)) h0n && 
             beq_nat h1n h0n &&
             beq_nat (S jn) h1n 
            then ok_arity jn 
            else error_Rec (ok_arity gn) (ok_arity h0n) 
              (ok_arity h1n) (ok_arity jn)
        | ag, ah0, ah1, aj => error_Rec ag ah0 ah1 aj 
      end
    | Comp n h l => 
      match arity h, map arity l with
        | ok_arity nh, al => 
          if beq_nat nh (length l) &&
            forallb (fun e => arity_eq e (ok_arity n)) al then
              ok_arity n else error_Comp (ok_arity nh) al
        | e , le => error_Comp e le 
      end
  end.

(* Specifications: *)
Lemma Cobham_ind_inf' : forall (P : nat -> Cobham -> Prop), forall Q : nat -> list Cobham -> Prop, (forall n, Q n nil) -> (forall e n l, P n e -> Q n l -> Q n (e :: l)) -> P 0 Zero -> (forall n i, i < n -> P n (Proj n i)) -> (forall b, P 1 (Succ b)) -> (P 2 Smash) -> (forall n g h0 h1 j, arity g = ok_arity n -> arity h0 = ok_arity (S (S n)) -> arity h1 = ok_arity (S (S n)) -> arity j = ok_arity (S n) -> P n g -> P (S (S n)) h0 -> P (S (S n)) h1 -> P (S n) j -> P (S n) (Rec g h0 h1 j)) -> (forall n h rl, arity h = ok_arity (length rl) -> (forall e, In e rl -> arity e = ok_arity n) -> P (length rl) h -> Q n rl -> P n (Comp n h rl)) -> forall e n , arity e = ok_arity n -> P n e.
Lemma Cobham_ind_inf : forall (P : nat -> Cobham -> Prop), P 0 Zero -> (forall n i, i < n -> P n (Proj n i)) -> (forall b, P 1 (Succ b)) -> (P 2 Smash) -> (forall n g h0 h1 j, arity g = ok_arity n -> arity h0 = ok_arity (S (S n)) -> arity h1 = ok_arity (S (S n)) -> arity j = ok_arity (S n) -> P n g -> P (S (S n)) h0 -> P (S (S n)) h1 -> P (S n) j -> P (S n) (Rec g h0 h1 j)) -> (forall n h rl, arity h = ok_arity (length rl) -> (forall e, In e rl -> arity e = ok_arity n) -> P (length rl) h -> (forall r, In r rl -> P n r) -> P n (Comp n h rl)) -> forall e n , arity e = ok_arity n -> P n e.