Imported Upstream version 8.1~gammaupstream/8.1.gamma

1 files changed, 31 insertions, 31 deletions

diff --git a/theories/ZArith/Zmax.v b/theories/ZArith/Zmax.v
index ae3bbf41..8af9b891 100644
--- a/theories/ZArith/Zmax.v
+++ b/theories/ZArith/Zmax.v
@@ -5,104 +5,104 @@
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
-(*i $Id: Zmax.v 8032 2006-02-12 21:20:48Z herbelin $ i*)
+(*i $Id: Zmax.v 9302 2006-10-27 21:21:17Z barras $ i*)
 
-Require Import Arith.
+Require Import Arith_base.
 Require Import BinInt.
 Require Import Zcompare.
 Require Import Zorder.
 
 Open Local Scope Z_scope.
 
-(**********************************************************************)
-(** *** Maximum of two binary integer numbers *)
+(******************************************)
+(** Maximum of two binary integer numbers *)
 
 Definition Zmax m n :=
-   match m ?= n with
+  match m ?= n with
     | Eq | Gt => m
     | Lt => n
-    end.
+  end.
 
-(** Characterization of maximum on binary integer numbers *)
+(** * Characterization of maximum on binary integer numbers *)
 
 Lemma Zmax_case : forall (n m:Z) (P:Z -> Type), P n -> P m -> P (Zmax n m).
 Proof.
-intros n m P H1 H2; unfold Zmax in |- *; case (n ?= m); auto with arith.
+  intros n m P H1 H2; unfold Zmax in |- *; case (n ?= m); auto with arith.
 Qed.
 
 Lemma Zmax_case_strong : forall (n m:Z) (P:Z -> Type), 
   (m<=n -> P n) -> (n<=m -> P m) -> P (Zmax n m).
 Proof.
-intros n m P H1 H2; unfold Zmax, Zle, Zge in *.
-rewrite <- (Zcompare_antisym n m) in H1.
-destruct (n ?= m); (apply H1|| apply H2); discriminate. 
+  intros n m P H1 H2; unfold Zmax, Zle, Zge in *.
+  rewrite <- (Zcompare_antisym n m) in H1.
+  destruct (n ?= m); (apply H1|| apply H2); discriminate. 
 Qed.
 
-(** Least upper bound properties of max *)
+(** * Least upper bound properties of max *)
 
 Lemma Zle_max_l : forall n m:Z, n <= Zmax n m.
 Proof.
-intros; apply Zmax_case_strong; auto with zarith.
+  intros; apply Zmax_case_strong; auto with zarith.
 Qed.
 
 Notation Zmax1 := Zle_max_l (only parsing).
 
 Lemma Zle_max_r : forall n m:Z, m <= Zmax n m.
 Proof.
-intros; apply Zmax_case_strong; auto with zarith.
+  intros; apply Zmax_case_strong; auto with zarith.
 Qed.
 
 Notation Zmax2 := Zle_max_r (only parsing).
 
 Lemma Zmax_lub : forall n m p:Z, n <= p -> m <= p -> Zmax n m <= p.
 Proof.
-intros; apply Zmax_case; assumption.
+  intros; apply Zmax_case; assumption.
 Qed.
 
-(** Semi-lattice properties of max *)
+(** * Semi-lattice properties of max *)
 
 Lemma Zmax_idempotent : forall n:Z, Zmax n n = n.
 Proof.
-intros; apply Zmax_case; auto.
+  intros; apply Zmax_case; auto.
 Qed.
 
 Lemma Zmax_comm : forall n m:Z, Zmax n m = Zmax m n.
 Proof.
-intros; do 2 apply Zmax_case_strong; intros; 
-  apply Zle_antisym; auto with zarith.
+  intros; do 2 apply Zmax_case_strong; intros; 
+    apply Zle_antisym; auto with zarith.
 Qed.
 
 Lemma Zmax_assoc : forall n m p:Z, Zmax n (Zmax m p) = Zmax (Zmax n m) p.
 Proof.
-intros n m p; repeat apply Zmax_case_strong; intros; 
-  reflexivity || (try apply Zle_antisym); eauto with zarith.
+  intros n m p; repeat apply Zmax_case_strong; intros; 
+    reflexivity || (try apply Zle_antisym); eauto with zarith.
 Qed.
 
-(** Additional properties of max *)
+(** * Additional properties of max *)
 
 Lemma Zmax_irreducible_inf : forall n m:Z, Zmax n m = n \/ Zmax n m = m.
 Proof.
-intros; apply Zmax_case; auto.
+  intros; apply Zmax_case; auto.
 Qed.
 
 Lemma Zmax_le_prime_inf : forall n m p:Z, p <= Zmax n m -> p <= n \/ p <= m.
 Proof.
-intros n m p; apply Zmax_case; auto.
+  intros n m p; apply Zmax_case; auto.
 Qed.
 
-(** Operations preserving max *)
+(** * Operations preserving max *)
 
 Lemma Zsucc_max_distr : 
   forall n m:Z, Zsucc (Zmax n m) = Zmax (Zsucc n) (Zsucc m).
 Proof.
-intros n m; unfold Zmax in |- *; rewrite (Zcompare_succ_compat n m);
- elim_compare n m; intros E; rewrite E; auto with arith.
+  intros n m; unfold Zmax in |- *; rewrite (Zcompare_succ_compat n m);
+    elim_compare n m; intros E; rewrite E; auto with arith.
 Qed.
 
 Lemma Zplus_max_distr_r : forall n m p:Z, Zmax (n + p) (m + p) = Zmax n m + p.
 Proof.
-intros x y n; unfold Zmax in |- *.
-rewrite (Zplus_comm x n); rewrite (Zplus_comm y n);
- rewrite (Zcompare_plus_compat x y n).
-case (x ?= y); apply Zplus_comm.
+  intros x y n; unfold Zmax 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.