Library Coq.Arith.Factorial


Require Import Plus.

Require Import Mult.

Require Import Lt.

Open Local Scope nat_scope.

Factorial





Boxed Fixpoint fact (n:nat) : nat :=

  match n with

    | O => 1

    | S n => S n * fact n

  end.




Arguments Scope fact [nat_scope].




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

Proof.

  simple induction n; unfold lt in |- *; simpl in |- *; auto with arith.

Qed.




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

Proof.

  intro.

  apply sym_not_eq.

  apply lt_O_neq.

  apply lt_O_fact.

Qed.




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

Proof.

  induction 1.

  apply le_n.

  assert (1 * fact n <= S m * fact m).

  apply mult_le_compat.

  apply lt_le_S; apply lt_O_Sn.

  assumption.

  simpl (1 * fact n) in H0.

  rewrite <- plus_n_O in H0.

  assumption.

Qed.