1 files changed, 106 insertions, 0 deletions
diff --git a/theories/ZArith/Zmin.v b/theories/ZArith/Zmin.v
new file mode 100644
index 00000000..d48e62c5
--- /dev/null
+++ b/theories/ZArith/Zmin.v
@@ -0,0 +1,106 @@
+(************************************************************************)
+(*  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 1.3.2.1 2004/07/16 19:31:21 herbelin Exp $ i*)
+
+(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *)
+
+Require Import Arith.
+Require Import BinInt.
+Require Import Zcompare.
+Require Import Zorder.
+
+Open Local Scope Z_scope.
+
+(**********************************************************************)
+(** Minimum on binary integer numbers *)
+
+Definition Zmin (n m:Z) :=
+  match n ?= m return Z with
+  | Eq => n
+  | Lt => n
+  | Gt => m
+  end.
+
+(** Properties of minimum on binary integer numbers *)
+
+Lemma Zmin_SS : forall n m:Z, Zsucc (Zmin n m) = Zmin (Zsucc n) (Zsucc m).
+Proof.
+intros n m; unfold Zmin in |- *; rewrite (Zcompare_succ_compat n m);
+ elim_compare n m; intros E; rewrite E; auto with arith.
+Qed.
+
+Lemma Zle_min_l : forall n m:Z, Zmin n m <= n.
+Proof.
+intros n m; unfold Zmin in |- *; elim_compare n m; intros E; rewrite E;
+ [ apply Zle_refl
+ | apply Zle_refl
+ | apply Zlt_le_weak; apply Zgt_lt; exact E ].
+Qed.
+
+Lemma Zle_min_r : forall n m:Z, Zmin n m <= m.
+Proof.
+intros n m; unfold Zmin in |- *; elim_compare n m; intros E; rewrite E;
+ [ unfold Zle in |- *; rewrite E; discriminate
+ | unfold Zle in |- *; rewrite E; discriminate
+ | apply Zle_refl ].
+Qed.
+
+Lemma Zmin_case : forall (n m:Z) (P:Z -> Set), P n -> P m -> P (Zmin n m).
+Proof.
+intros n m P H1 H2; unfold Zmin in |- *; case (n ?= m); auto with arith.
+Qed.
+
+Lemma Zmin_or : forall n m:Z, Zmin n m = n \/ Zmin n m = m.
+Proof.
+unfold Zmin in |- *; intros; elim (n ?= m); auto.
+Qed.
+
+Lemma Zmin_n_n : forall n:Z, Zmin n n = n.
+Proof.
+unfold Zmin in |- *; intros; elim (n ?= n); auto.
+Qed.
+
+Lemma Zmin_plus : forall n m p:Z, Zmin (n + p) (m + p) = Zmin n m + p.
+Proof.
+intros x y n; unfold Zmin in |- *.
+rewrite (Zplus_comm x n); rewrite (Zplus_comm y n);
+ rewrite (Zcompare_plus_compat x y n).
+case (x ?= y); apply Zplus_comm.
+Qed.
+
+(**********************************************************************)
+(** Maximum of two binary integer numbers *)
+
+Definition Zmax a b := match a ?= b with
+                       | Lt => b
+                       | _ => a
+                       end.
+
+(** Properties of maximum on binary integer numbers *)
+
+Ltac CaseEq name :=
+  generalize (refl_equal name); pattern name at -1 in |- *; case name.
+
+Theorem Zmax1 : forall a b, a <= Zmax a b.
+Proof.
+intros a b; unfold Zmax in |- *; CaseEq (a ?= b); simpl in |- *;
+ auto with zarith.
+unfold Zle in |- *; intros H; rewrite H; red in |- *; intros; discriminate.
+Qed.
+
+Theorem Zmax2 : forall a b, b <= Zmax a b.
+Proof.
+intros a b; unfold Zmax in |- *; CaseEq (a ?= b); simpl in |- *;
+ auto with zarith.
+intros H;
+ (case (Zle_or_lt b a); auto; unfold Zlt in |- *; rewrite H; intros;
+   discriminate).
+intros H;
+ (case (Zle_or_lt b a); auto; unfold Zlt in |- *; rewrite H; intros;
+   discriminate).
+Qed.