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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Factorial.v 13323 2010-07-24 15:57:30Z herbelin $ i*)
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.
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