From 6b649aba925b6f7462da07599fe67ebb12a3460e Mon Sep 17 00:00:00 2001
From: Samuel Mimram <samuel.mimram@ens-lyon.org>
Date: Wed, 28 Jul 2004 21:54:47 +0000
Subject: Imported Upstream version 8.0pl1

---
 theories/Arith/Min.v | 83 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 83 insertions(+)
 create mode 100755 theories/Arith/Min.v

(limited to 'theories/Arith/Min.v')

diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v
new file mode 100755
index 00000000..912e7ba3
--- /dev/null
+++ b/theories/Arith/Min.v
@@ -0,0 +1,83 @@
+(************************************************************************)
+(*  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: Min.v,v 1.10.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
+
+Require Import Arith.
+
+Open Local Scope nat_scope.
+
+Implicit Types m n : nat.
+
+(** minimum of two natural numbers *)
+
+Fixpoint min n m {struct n} : nat :=
+  match n, m with
+  | O, _ => 0
+  | S n', O => 0
+  | S n', S m' => S (min n' m')
+  end.
+
+(** Simplifications of [min] *)
+
+Lemma min_SS : forall n m, S (min n m) = min (S n) (S m).
+Proof.
+auto with arith.
+Qed.
+
+Lemma min_comm : forall n m, min n m = min m n.
+Proof.
+induction n; induction m; simpl in |- *; auto with arith.
+Qed.
+
+(** [min] and [le] *)
+
+Lemma min_l : forall n m, n <= m -> min n m = n.
+Proof.
+induction n; induction m; simpl in |- *; auto with arith.
+Qed.
+
+Lemma min_r : forall n m, m <= n -> min n m = m.
+Proof.
+induction n; induction m; simpl in |- *; auto with arith.
+Qed.
+
+Lemma le_min_l : forall n m, min n m <= n.
+Proof.
+induction n; intros; simpl in |- *; auto with arith.
+elim m; intros; simpl in |- *; auto with arith.
+Qed.
+
+Lemma le_min_r : forall n m, min n m <= m.
+Proof.
+induction n; simpl in |- *; auto with arith.
+induction m; simpl in |- *; auto with arith.
+Qed.
+Hint Resolve min_l min_r le_min_l le_min_r: arith v62.
+
+(** [min n m] is equal to [n] or [m] *)
+
+Lemma min_dec : forall n m, {min n m = n} + {min n m = m}.
+Proof.
+induction n; induction m; simpl in |- *; auto with arith.
+elim (IHn m); intro H; elim H; auto.
+Qed.
+
+Lemma min_case : forall n m (P:nat -> Set), P n -> P m -> P (min n m).
+Proof.
+induction n; simpl in |- *; auto with arith.
+induction m; intros; simpl in |- *; auto with arith.
+pattern (min n m) in |- *; apply IHn; auto with arith.
+Qed.
+
+Lemma min_case2 : forall n m (P:nat -> Prop), P n -> P m -> P (min n m).
+Proof.
+induction n; simpl in |- *; auto with arith.
+induction m; intros; simpl in |- *; auto with arith.
+pattern (min n m) in |- *; apply IHn; auto with arith.
+Qed.
\ No newline at end of file
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