Deplacement des fichiers ancienne syntaxe dans theories7, contrib7 et states7

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@5026 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 51 insertions, 0 deletions

diff --git a/theories7/Arith/Factorial.v b/theories7/Arith/Factorial.v
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+(***********************************************************************)
+(*  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 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.