Numbers: properties of min/max with respect to 0,S,P,add,sub,mul

 With these properties, we can kill Arith/MinMax, NArith/Nminmax,
 and leave ZArith/Zminmax as a compatibility file only. Now
 the instanciations NPeano.Nat, NBinary.N, ZBinary.Z, BigZ, BigN
 contains all theses facts.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12718 85f007b7-540e-0410-9357-904b9bb8a0f7

4 files changed, 20 insertions, 79 deletions

diff --git a/theories/Arith/Max.v b/theories/Arith/Max.v
index e49251a71..036b47ba2 100644
--- a/theories/Arith/Max.v
+++ b/theories/Arith/Max.v
@@ -8,21 +8,20 @@
 
 (*i $Id$ i*)
 
-(** THIS FILE IS DEPRECATED. Use [NPeano] and [MinMax] instead. *)
+(** THIS FILE IS DEPRECATED. Use [NPeano.Nat] instead. *)
 
 Require Import NPeano.
-Require Export MinMax.
 
 Local Open Scope nat_scope.
 Implicit Types m n p : nat.
 
 Notation max := NPeano.max (only parsing).
 
-Definition max_0_l := max_0_l.
-Definition max_0_r := max_0_r.
-Definition succ_max_distr := succ_max_distr.
-Definition plus_max_distr_l := plus_max_distr_l.
-Definition plus_max_distr_r := plus_max_distr_r.
+Definition max_0_l := Nat.max_0_l.
+Definition max_0_r := Nat.max_0_r.
+Definition succ_max_distr := Nat.succ_max_distr.
+Definition plus_max_distr_l := Nat.add_max_distr_l.
+Definition plus_max_distr_r := Nat.add_max_distr_r.
 Definition max_case_strong := Nat.max_case_strong.
 Definition max_spec := Nat.max_spec.
 Definition max_dec := Nat.max_dec.
@@ -41,5 +40,11 @@ Definition max_lub := Nat.max_lub.
 (* begin hide *)
 (* Compatibility *)
 Notation max_case2 := max_case (only parsing).
-Notation max_SS := succ_max_distr (only parsing).
+Notation max_SS := Nat.succ_max_distr (only parsing).
 (* end hide *)
+
+Hint Resolve
+ Nat.max_l Nat.max_r Nat.le_max_l Nat.le_max_r : arith v62.
+
+Hint Resolve
+ Nat.min_l Nat.min_r Nat.le_min_l Nat.le_min_r : arith v62.
diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v
index 44060b56c..521285a08 100644
--- a/theories/Arith/Min.v
+++ b/theories/Arith/Min.v
@@ -8,21 +8,20 @@
 
 (*i $Id$ i*)
 
-(** THIS FILE IS DEPRECATED. Use [NPeano] and [MinMax] instead. *)
+(** THIS FILE IS DEPRECATED. Use [NPeano.Nat] instead. *)
 
 Require Import NPeano.
-Require Export MinMax.
 
 Open Local Scope nat_scope.
 Implicit Types m n p : nat.
 
 Notation min := NPeano.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_0_l := Nat.min_0_l.
+Definition min_0_r := Nat.min_0_r.
+Definition succ_min_distr := Nat.succ_min_distr.
+Definition plus_min_distr_l := Nat.add_min_distr_l.
+Definition plus_min_distr_r := Nat.add_min_distr_r.
 Definition min_case_strong := Nat.min_case_strong.
 Definition min_spec := Nat.min_spec.
 Definition min_dec := Nat.min_dec.
@@ -41,5 +40,5 @@ Definition min_glb := Nat.min_glb.
 (* begin hide *)
 (* Compatibility *)
 Notation min_case2 := min_case (only parsing).
-Notation min_SS := succ_min_distr (only parsing).
+Notation min_SS := Nat.succ_min_distr (only parsing).
 (* end hide *)
\ No newline at end of file
diff --git a/theories/Arith/MinMax.v b/theories/Arith/MinMax.v
deleted file mode 100644
index 9ce43ab8c..000000000
--- a/theories/Arith/MinMax.v
+++ /dev/null
@@ -1,62 +0,0 @@
-(************************************************************************)
-(*  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        *)
-(************************************************************************)
-
-Require Import Orders NPeano.
-
-(** The Definition of maximum and minimum of two natural numbers
-    is now in NPeano, as well as generic properties. *)
-
-(** * Properties specific to the [nat] domain *)
-
-(** Simplifications *)
-
-Lemma max_0_l : forall n, max 0 n = n.
-Proof. reflexivity. Qed.
-
-Lemma max_0_r : forall n, max n 0 = n.
-Proof. destruct n; auto. Qed.
-
-Lemma min_0_l : forall n, min 0 n = 0.
-Proof. reflexivity. Qed.
-
-Lemma min_0_r : forall n, min n 0 = 0.
-Proof. destruct n; auto. Qed.
-
-(** Compatibilities (consequences of monotonicity) *)
-
-Lemma succ_max_distr : forall n m, S (max n m) = max (S n) (S m).
-Proof. auto. Qed.
-
-Lemma succ_min_distr : forall n m, S (min n m) = min (S n) (S m).
-Proof. auto. Qed.
-
-Lemma plus_max_distr_l : forall n m p, max (p + n) (p + m) = p + max n m.
-Proof.
-intros. apply Nat.max_monotone. repeat red; auto with arith.
-Qed.
-
-Lemma plus_max_distr_r : forall n m p, max (n + p) (m + p) = max n m + p.
-Proof.
-intros. apply Nat.max_monotone with (f:=fun x => x + p).
-repeat red; auto with arith.
-Qed.
-
-Lemma plus_min_distr_l : forall n m p, min (p + n) (p + m) = p + min n m.
-Proof.
-intros. apply Nat.min_monotone. repeat red; auto with arith.
-Qed.
-
-Lemma plus_min_distr_r : forall n m p, min (n + p) (m + p) = min n m + p.
-Proof.
-intros. apply Nat.min_monotone with (f:=fun x => x + p).
-repeat red; auto with arith.
-Qed.
-
-Hint Resolve
- Nat.max_l Nat.max_r Nat.le_max_l Nat.le_max_r
- Nat.min_l Nat.min_r Nat.le_min_l Nat.le_min_r : arith v62.
diff --git a/theories/Arith/vo.itarget b/theories/Arith/vo.itarget
index d8bff0ab9..0b6564e15 100644
--- a/theories/Arith/vo.itarget
+++ b/theories/Arith/vo.itarget
@@ -19,4 +19,3 @@ Mult.vo
 Peano_dec.vo
 Plus.vo
 Wf_nat.vo
-MinMax.vo