Fixpoint firstIndexOf(A : Set)(eqd : eq_dec A)(ls : list A)(a : A)(def : nat) :=
  match ls with
      | nil => def
      | a' :: ls' =>
        if (eqd a a') then O else (S (firstIndexOf eqd ls' a def))
  end.

Theorem firstIndexOf_in_lt : forall (A : Set)(eqd : eq_dec A)(ls : list A)(a : A)(def : nat), In a ls -> firstIndexOf eqd ls a def < length ls.