Nouveaux lemmes (sur proposition de Nijmegen)

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@4012 85f007b7-540e-0410-9357-904b9bb8a0f7

2 files changed, 35 insertions, 0 deletions

diff --git a/theories/Arith/Arith.v b/theories/Arith/Arith.v
index 13c17aabb5..832ea7a427 100755
--- a/theories/Arith/Arith.v
+++ b/theories/Arith/Arith.v
@@ -18,3 +18,4 @@ Require Export Between.
 Require Export Minus.
 Require Export Peano_dec.
 Require Export Compare_dec.
+Require Export Factorial.
diff --git a/theories/Arith/Factorial.v b/theories/Arith/Factorial.v
new file mode 100644
index 0000000000..4963379e68
--- /dev/null
+++ b/theories/Arith/Factorial.v
@@ -0,0 +1,34 @@
+(***********************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
+(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
+(*   \VV/  *************************************************************)
+(*    //   *      This file is distributed under the terms of the      *)
+(*         *       GNU Lesser General Public License Version 2.1       *)
+(***********************************************************************)
+
+(*i $Id$ i*)
+
+Require Plus.
+Require Lt.
+
+(** Factorial *)
+
+Fixpoint fact [n:nat]:nat:=
+  Cases n of
+     O     => (S O)
+    |(S n) => (mult (S n) (fact n))
+  end.
+
+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.
+