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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: Zmin.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*)
+
+(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *)
+
+Require Arith.
+Require BinInt.
+Require Zcompare.
+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 x y n; 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.
+
+(**********************************************************************)
+(** Maximum of two binary integer numbers *)
+V7only [ (* From Zdivides *) ].
+
+Definition Zmax :=
+   [a, b : ?]  Cases (Zcompare a b) of INFERIEUR => b | _ => a end.
+
+(** Properties of maximum on binary integer numbers *)
+
+Tactic Definition CaseEq name :=
+Generalize (refl_equal ? name); Pattern -1 name; Case name.
+
+Theorem Zmax1: (a, b : ?)  (Zle a (Zmax a b)).
+Proof.
+Intros a b; Unfold Zmax; (CaseEq '(Zcompare a b)); Simpl; Auto with zarith.
+Unfold Zle; Intros H; Rewrite H; Red; Intros; Discriminate.
+Qed.
+
+Theorem Zmax2: (a, b : ?)  (Zle b (Zmax a b)).
+Proof.
+Intros a b; Unfold Zmax; (CaseEq '(Zcompare a b)); Simpl; Auto with zarith.
+Intros H;
+ (Case (Zle_or_lt b a); Auto; Unfold Zlt; Rewrite H; Intros; Discriminate).
+Intros H;
+ (Case (Zle_or_lt b a); Auto; Unfold Zlt; Rewrite H; Intros; Discriminate).
+Qed.
+