Library Coq.Arith.Factorial


Require Import Plus.

Require Import Mult.

Require Import Lt.

Open Local Scope nat_scope.

Factorial





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.

Index

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