Restructuration ZArith et déport de la partie sur 'positive' dans NArith, de la partie Omega dans contrib/omega

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@4801 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 74 insertions, 0 deletions

diff --git a/theories/ZArith/Zmin.v b/theories/ZArith/Zmin.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*)
+
+(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *)
+
+Require Arith.
+Require fast_integer.
+Require Zorder.
+
+Open Local Scope Z_scope.
+
+(**********************************************************************)
+(** Minimum on binary integer numbers *)
+
+Definition Zmin := [n,m:Z]
+ <Z>Cases (Zcompare n m) of
+      EGAL      => n
+    | INFERIEUR => n
+    | SUPERIEUR => m
+    end.
+
+(** Properties of minimum on binary integer numbers *)
+
+Lemma Zmin_SS : (n,m:Z)((Zs (Zmin n m))=(Zmin (Zs n) (Zs m))).
+Proof.
+Intros n m;Unfold Zmin; Rewrite (Zcompare_n_S n m);
+(ElimCompare 'n 'm);Intros E;Rewrite E;Auto with arith.
+Qed.
+
+Lemma Zle_min_l : (n,m:Z)(Zle (Zmin n m) n).
+Proof.
+Intros n m;Unfold Zmin ; (ElimCompare 'n 'm);Intros E;Rewrite -> E;
+  [ Apply Zle_n | Apply Zle_n | Apply Zlt_le_weak; Apply Zgt_lt;Exact E ].
+Qed.
+
+Lemma Zle_min_r : (n,m:Z)(Zle (Zmin n m) m).
+Proof.
+Intros n m;Unfold Zmin ; (ElimCompare 'n 'm);Intros E;Rewrite -> E;[
+  Unfold Zle ;Rewrite -> E;Discriminate
+| Unfold Zle ;Rewrite -> E;Discriminate
+| Apply Zle_n ].
+Qed.
+
+Lemma Zmin_case : (n,m:Z)(P:Z->Set)(P n)->(P m)->(P (Zmin n m)).
+Proof.
+Intros n m P H1 H2; Unfold Zmin; Case (Zcompare n m);Auto with arith.
+Qed.
+
+Lemma Zmin_or : (n,m:Z)(Zmin n m)=n \/ (Zmin n m)=m.
+Proof.
+Unfold Zmin; Intros; Elim (Zcompare n m); Auto.
+Qed.
+
+Lemma Zmin_n_n : (n:Z) (Zmin n n)=n.
+Proof.
+Unfold Zmin; Intros; Elim (Zcompare n n); Auto.
+Qed.
+
+Lemma Zmin_plus :
+ (x,y,n:Z)(Zmin (Zplus x n) (Zplus y n))=(Zplus (Zmin x y) n).
+Proof.
+Intros; Unfold Zmin.
+Rewrite (Zplus_sym x n);
+Rewrite (Zplus_sym y n);
+Rewrite (Zcompare_Zplus_compatible x y n).
+Case (Zcompare x y); Apply Zplus_sym.
+Qed.
+