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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id: Factorial.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*)
+
+Require Plus.
+Require Mult.
+Require Lt.
+V7only [Import nat_scope.].
+Open Local Scope nat_scope.
+
+(** Factorial *)
+
+Fixpoint fact [n:nat]:nat:=
+  Cases n of
+     O     => (S O)
+    |(S n) => (mult (S n) (fact n))
+  end.
+
+Arguments Scope fact [ nat_scope ].
+
+Lemma lt_O_fact : (n:nat)(lt O (fact n)).
+Proof.
+Induction n; Unfold lt; Simpl; Auto with arith.
+Qed.
+
+Lemma fact_neq_0:(n:nat)~(fact n)=O.
+Proof.
+Intro.
+Apply sym_not_eq.
+Apply lt_O_neq.
+Apply lt_O_fact.
+Qed.
+
+Lemma fact_growing : (n,m:nat) (le n m) -> (le (fact n) (fact m)).
+Proof.
+NewInduction 1.
+Apply le_n.
+Assert (le (mult (S O) (fact n)) (mult (S m) (fact m))).
+Apply le_mult_mult.
+Apply lt_le_S; Apply lt_O_Sn.
+Assumption.
+Simpl (mult (S O) (fact n)) in H0.
+Rewrite <- plus_n_O in H0.
+Assumption.
+Qed.