Definition falling_factorial := nosimpl ffact_rec.Notation "''C' ( n , m )" := (binomial n m)
  (at level 8, format "''C' ( n ,  m )") : nat_scope.

(* Specifications: *)
Lemma ffactE : falling_factorial = ffact_rec.
Lemma ffactn0 n : n ^_ 0 = 1.
Lemma ffactn1 n : n ^_ 1 = n.
Lemma ffact_gt0 n m : (0 < n ^_ m) = (m <= n).
Lemma ffact_small n m : n < m -> n ^_ m = 0.