Library Coq.Arith.Factorial

Require Import PeanoNat Plus Mult Lt.
Local Open Scope nat_scope.

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

Arguments fact n%nat.

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.