Ajout d'un fichier Max dans Arith, et enrichissement du Min.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2195 85f007b7-540e-0410-9357-904b9bb8a0f7

2 files changed, 118 insertions, 2 deletions

diff --git a/theories/Arith/Max.v b/theories/Arith/Max.v
new file mode 100755
index 0000000000..5ef1ba4456
--- /dev/null
+++ b/theories/Arith/Max.v
@@ -0,0 +1,85 @@
+(***********************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
+(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
+(*   \VV/  *************************************************************)
+(*    //   *      This file is distributed under the terms of the      *)
+(*         *       GNU Lesser General Public License Version 2.1       *)
+(***********************************************************************)
+
+(*i $Id$ *)
+
+Require Arith.
+
+(**************************************************************************)
+(*                    maximum of two natural numbers                      *)
+(**************************************************************************)
+
+Fixpoint max [n:nat] : nat -> nat :=  
+[m:nat]Cases n m of
+          O      _     => m
+       | (S n')  O     => n
+       | (S n') (S m') => (S (max n' m'))
+       end.
+
+(* Simplifications of max *)
+
+
+Lemma max_SS : (n,m:nat)((S (max n m))=(max (S n) (S m))).
+Proof.
+Auto with arith.
+Qed.
+
+Lemma max_sym : (n,m:nat)(max n m)=(max m n).
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+(* max and le *)
+
+Lemma max_l : (n,m:nat)(le m n)->(max n m)=n.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+Lemma max_r : (n,m:nat)(le n m)->(max n m)=m.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+Lemma le_max_l : (n,m:nat)(le n (max n m)).
+Proof.
+NewInduction n; Intros; Simpl; Auto with arith.
+Elim m; Intros; Simpl; Auto with arith.
+Qed.
+
+Lemma le_max_r : (n,m:nat)(le m (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Simpl; Auto with arith.
+Qed.
+Hints Resolve max_r max_l le_max_l le_max_r: arith v62.
+
+
+(* max n m is equal to n or m *)
+
+Lemma max_dec : (n,m:nat){(max n m)=n}+{(max n m)=m}.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Elim (IHn m);Intro H;Elim H;Auto.
+Qed.
+
+Lemma max_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Intros; Simpl; Auto with arith.
+Pattern (max n m); Apply IHn ; Auto with arith.
+Qed.
+
+Lemma max_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Intros; Simpl; Auto with arith.
+Pattern (max n m); Apply IHn ; Auto with arith.
+Qed.
+
+
diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v
index a38329c347..56d254c481 100755
--- a/theories/Arith/Min.v
+++ b/theories/Arith/Min.v
@@ -21,30 +21,61 @@ Fixpoint min [n:nat] : nat -> nat :=
        | (S n') (S m') => (S (min n' m'))
        end.
 
+(* Simplifications of min *)
+
 Lemma min_SS : (n,m:nat)((S (min n m))=(min (S n) (S m))).
 Proof.
 Auto with arith.
 Qed.
 
+Lemma min_sym : (n,m:nat)(min n m)=(min m n).
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+(* min and le *)
+
+Lemma min_l : (n,m:nat)(le n m)->(min n m)=n.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+Lemma min_r : (n,m:nat)(le m n)->(min n m)=m.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
 Lemma le_min_l : (n,m:nat)(le (min n m) n).
 Proof.
 NewInduction n; Intros; Simpl; Auto with arith.
 Elim m; Intros; Simpl; Auto with arith.
 Qed.
-Hints Resolve le_min_l : arith v62.
 
 Lemma le_min_r : (n,m:nat)(le (min n m) m).
 Proof.
 NewInduction n; Simpl; Auto with arith.
 NewInduction m; Simpl; Auto with arith.
 Qed.
-Hints Resolve le_min_r : arith v62.
+Hints 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 : (n,m:nat){(min n m)=n}+{(min n m)=m}.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Elim (IHn m);Intro H;Elim H;Auto.
+Qed.
+
 Lemma min_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (min n m)).
 Proof.
 NewInduction n; Simpl; Auto with arith.
 NewInduction m; Intros; Simpl; Auto with arith.
 Pattern (min n m); Apply IHn ; Auto with arith.
 Qed.
+
+Lemma min_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (min n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Intros; Simpl; Auto with arith.
+Pattern (min n m); Apply IHn ; Auto with arith.
+Qed.