Nettoyage Zmin.v, création Zmax.v et Zminmax.v

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@8032 85f007b7-540e-0410-9357-904b9bb8a0f7

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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 $Id$ i*)
+
+Require Import Zmin Zmax.
+Require Import BinInt Zorder.
+
+Open Scope Z_scope.
+
+(** *** Lattice properties of min and max on Z *)
+
+(** Absorption *)
+
+Lemma Zmin_max_absorption_r_r : forall n m, Zmax n (Zmin n m) = n.
+Proof.
+intros; apply Zmin_case_strong; intro; apply Zmax_case_strong; intro; 
+  reflexivity || apply Zle_antisym; trivial.
+Qed.
+
+Lemma Zmax_min_absorption_r_r : forall n m, Zmin n (Zmax n m) = n.
+Proof.
+intros; apply Zmax_case_strong; intro; apply Zmin_case_strong; intro; 
+  reflexivity || apply Zle_antisym; trivial.
+Qed.
+
+(** Distributivity *)
+
+Lemma Zmax_min_distr_r : 
+  forall n m p, Zmax n (Zmin m p) = Zmin (Zmax n m) (Zmax n p).
+Proof.
+intros.
+repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros; 
+  reflexivity ||
+  apply Zle_antisym; (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+Lemma Zmin_max_distr_r : 
+  forall n m p, Zmin n (Zmax m p) = Zmax (Zmin n m) (Zmin n p).
+Proof.
+intros.
+repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros; 
+  reflexivity ||
+  apply Zle_antisym; (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+(** Modularity *)
+
+Lemma Zmax_min_modular_r :
+  forall n m p, Zmax n (Zmin m (Zmax n p)) = Zmin (Zmax n m) (Zmax n p).
+Proof.
+intros; repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros;
+  reflexivity ||
+  apply Zle_antisym; (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+Lemma Zmin_max_modular_r :
+  forall n m p, Zmin n (Zmax m (Zmin n p)) = Zmax (Zmin n m) (Zmin n p).
+Proof.
+intros; repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros;
+  reflexivity ||
+  apply Zle_antisym; (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+(** Disassociativity *)
+
+Lemma max_min_disassoc : forall n m p, Zmin n (Zmax m p) <= Zmax (Zmin n m) p.
+Proof.
+intros; repeat apply Zmax_case_strong; repeat apply Zmin_case_strong; intros;
+  apply Zle_refl || (assumption || eapply Zle_trans; eassumption).
+Qed.
+
+
+
+
+
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