Ajout d'un fichier Max dans Arith, et enrichissement du Min.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2195 85f007b7-540e-0410-9357-904b9bb8a0f7

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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 $Id$ *)
+
+Require Arith.
+
+(**************************************************************************)
+(*                    maximum of two natural numbers                      *)
+(**************************************************************************)
+
+Fixpoint max [n:nat] : nat -> nat :=  
+[m:nat]Cases n m of
+          O      _     => m
+       | (S n')  O     => n
+       | (S n') (S m') => (S (max n' m'))
+       end.
+
+(* Simplifications of max *)
+
+
+Lemma max_SS : (n,m:nat)((S (max n m))=(max (S n) (S m))).
+Proof.
+Auto with arith.
+Qed.
+
+Lemma max_sym : (n,m:nat)(max n m)=(max m n).
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+(* max and le *)
+
+Lemma max_l : (n,m:nat)(le m n)->(max n m)=n.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+Lemma max_r : (n,m:nat)(le n m)->(max n m)=m.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+Lemma le_max_l : (n,m:nat)(le n (max n m)).
+Proof.
+NewInduction n; Intros; Simpl; Auto with arith.
+Elim m; Intros; Simpl; Auto with arith.
+Qed.
+
+Lemma le_max_r : (n,m:nat)(le m (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Simpl; Auto with arith.
+Qed.
+Hints Resolve max_r max_l le_max_l le_max_r: arith v62.
+
+
+(* max n m is equal to n or m *)
+
+Lemma max_dec : (n,m:nat){(max n m)=n}+{(max n m)=m}.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Elim (IHn m);Intro H;Elim H;Auto.
+Qed.
+
+Lemma max_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Intros; Simpl; Auto with arith.
+Pattern (max n m); Apply IHn ; Auto with arith.
+Qed.
+
+Lemma max_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Intros; Simpl; Auto with arith.
+Pattern (max n m); Apply IHn ; Auto with arith.
+Qed.
+
+