Modularization of BinPos + fixes in Stdlib

 BinPos now contain a sub-module Pos, in which are placed functions
 like add (ex-Pplus), mul (ex-Pmult), ... and properties like
 add_comm, add_assoc, ...

 In addition to the name changes, the organisation is changed quite
 a lot, to try to take advantage more of the orders < and <= instead
 of speaking only of the comparison function.

 The main source of incompatibilities in scripts concerns this compare:
 Pos.compare is now a binary operation, expressed in terms of the
 ex-Pcompare which is ternary (expecting an initial comparision as 3rd arg),
 this ternary version being called now Pos.compare_cont. As for everything
 else, compatibility notations (only parsing) are provided. But notations
 "_ ?= _" on positive will have to be edited, since they now point to
 Pos.compare.

 We also make the sub-module Pos to be directly an OrderedType,
 and include results about min and max.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14098 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 7 insertions, 7 deletions

diff --git a/theories/NArith/Ngcd_def.v b/theories/NArith/Ngcd_def.v
index fe5185c6b..c25af8717 100644
--- a/theories/NArith/Ngcd_def.v
+++ b/theories/NArith/Ngcd_def.v
@@ -6,7 +6,7 @@
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
 
-Require Import BinPos BinNat Pgcd.
+Require Import BinPos BinNat.
 
 Local Open Scope N_scope.
 
@@ -21,7 +21,7 @@ Definition Ngcd (a b : N) :=
  match a, b with
   | 0, _ => b
   | _, 0 => a
-  | Npos p, Npos q => Npos (Pgcd p q)
+  | Npos p, Npos q => Npos (Pos.gcd p q)
  end.
 
 (** * Generalized Gcd, also computing rests of a and b after
@@ -32,7 +32,7 @@ Definition Nggcd (a b : N) :=
   | 0, _ => (b,(0,1))
   | _, 0 => (a,(1,0))
   | Npos p, Npos q =>
-     let '(g,(aa,bb)) := Pggcd p q in
+     let '(g,(aa,bb)) := Pos.ggcd p q in
      (Npos g, (Npos aa, Npos bb))
  end.
 
@@ -41,7 +41,7 @@ Definition Nggcd (a b : N) :=
 Lemma Nggcd_gcd : forall  a b, fst (Nggcd a b) = Ngcd a b.
 Proof.
  intros [ |p] [ |q]; simpl; auto.
- generalize (Pggcd_gcd p q). destruct Pggcd as (g,(aa,bb)); simpl; congruence.
+ generalize (Pos.ggcd_gcd p q). destruct Pos.ggcd as (g,(aa,bb)); simpl; congruence.
 Qed.
 
 (** The other components of Nggcd are indeed the correct factors. *)
@@ -53,8 +53,8 @@ Proof.
  intros [ |p] [ |q]; simpl; auto.
  now rewrite Pmult_1_r.
  now rewrite Pmult_1_r.
- generalize (Pggcd_correct_divisors p q).
- destruct Pggcd as (g,(aa,bb)); simpl. destruct 1; split; congruence.
+ generalize (Pos.ggcd_correct_divisors p q).
+ destruct Pos.ggcd as (g,(aa,bb)); simpl. destruct 1; split; congruence.
 Qed.
 
 (** We can use this fact to prove a part of the gcd correctness *)
@@ -78,7 +78,7 @@ Proof.
  intros [ |p] [ |q]; simpl; auto.
  intros [ |r]. intros (s,H). discriminate.
  intros ([ |s],Hs) ([ |t],Ht); try discriminate; simpl in *.
- destruct (Pgcd_greatest p q r) as (u,H).
+ destruct (Pos.gcd_greatest p q r) as (u,H).
  exists s. now inversion Hs.
  exists t. now inversion Ht.
  exists (Npos u). simpl; now f_equal.