(** Factorial *)

Fixpoint fact (n:nat) : nat :=
  match n with
    | O => 1
    | S n => S n * fact n
  end.

(* Specifications: *)
Lemma lt_O_fact n : 0 < fact n.
Lemma fact_neq_0 n : fact n <> 0.
Lemma fact_le n m : n <= m -> fact n <= fact m.