From 5b7eafd0f00a16d78f99a27f5c7d5a0de77dc7e6 Mon Sep 17 00:00:00 2001
From: Stephane Glondu <steph@glondu.net>
Date: Wed, 21 Jul 2010 09:46:51 +0200
Subject: Imported Upstream snapshot 8.3~beta0+13298

---
 theories/Arith/Min.v | 116 ++++++++++++++-------------------------------------
 1 file changed, 32 insertions(+), 84 deletions(-)

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

diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v
index aa009963..c52fc0dd 100644
--- a/theories/Arith/Min.v
+++ b/theories/Arith/Min.v
@@ -6,91 +6,39 @@
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
 
-(*i $Id: Min.v 9660 2007-02-19 11:36:30Z notin $ i*)
+(*i $Id$ i*)
 
-Require Import Le.
+(** THIS FILE IS DEPRECATED. Use [MinMax] instead. *)
 
-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_0_l : forall n : nat, min 0 n = 0.
-Proof.
-  trivial. 
-Qed.
-
-Lemma min_0_r : forall n : nat, min n 0 = 0.
-Proof.
-  destruct n; trivial.
-Qed.
-
-Lemma min_SS : forall n m, S (min n m) = min (S n) (S m).
-Proof.
-  auto with arith.
-Qed.
-
-Lemma min_assoc : forall m n p : nat, min m (min n p) = min (min m n) p.
-Proof.
-  induction m; destruct n; destruct p; trivial.
-  simpl.
-  auto using (IHm n p).
-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 -> Type), 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.
+Require Export MinMax.
 
+Open Local Scope nat_scope.
+Implicit Types m n p : nat.
+
+Notation min := MinMax.min (only parsing).
+
+Definition min_0_l := min_0_l.
+Definition min_0_r := min_0_r.
+Definition succ_min_distr := succ_min_distr.
+Definition plus_min_distr_l := plus_min_distr_l.
+Definition plus_min_distr_r := plus_min_distr_r.
+Definition min_case_strong := min_case_strong.
+Definition min_spec := min_spec.
+Definition min_dec := min_dec.
+Definition min_case := min_case.
+Definition min_idempotent := min_id.
+Definition min_assoc := min_assoc.
+Definition min_comm := min_comm.
+Definition min_l := min_l.
+Definition min_r := min_r.
+Definition le_min_l := le_min_l.
+Definition le_min_r := le_min_r.
+Definition min_glb_l := min_glb_l.
+Definition min_glb_r := min_glb_r.
+Definition min_glb := min_glb.
+
+(* begin hide *)
+(* Compatibility *)
 Notation min_case2 := min_case (only parsing).
-
+Notation min_SS := succ_min_distr (only parsing).
+(* end hide *)
\ No newline at end of file
-- 
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