Library Coq.Arith.Factorial

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

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


Lemma lt_O_fact : forall n:nat, 0 < fact n.

Lemma fact_neq_0 : forall n:nat, fact n <> 0.

Lemma fact_le : forall n m:nat, n <= m -> fact n <= fact m.