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, 6 insertions, 8 deletions

diff --git a/theories/NArith/Ndiv_def.v b/theories/NArith/Ndiv_def.v
index 2a3fd152a..0850a631e 100644
--- a/theories/NArith/Ndiv_def.v
+++ b/theories/NArith/Ndiv_def.v
@@ -14,7 +14,7 @@ Local Open Scope N_scope.
 Definition NPgeb (a:N)(b:positive) :=
   match a with
    | 0 => false
-   | Npos na => match Pcompare na b Eq with Lt => false | _ => true end
+   | Npos na => match Pos.compare na b with Lt => false | _ => true end
   end.
 
 Local Notation "a >=? b" := (NPgeb a b) (at level 70).
@@ -54,24 +54,22 @@ Lemma NPgeb_ge : forall a b, NPgeb a b = true -> a >= Npos b.
 Proof.
  destruct a; simpl; intros.
  discriminate.
- unfold Nge, Ncompare. now destruct Pcompare.
+ unfold Nge, Ncompare. now destruct Pos.compare.
 Qed.
 
 Lemma NPgeb_lt : forall a b, NPgeb a b = false -> a < Npos b.
 Proof.
  destruct a; simpl; intros. red; auto.
- unfold Nlt, Ncompare. now destruct Pcompare.
+ unfold Nlt, Ncompare. now destruct Pos.compare.
 Qed.
 
 Theorem NPgeb_correct: forall (a:N)(b:positive),
   if NPgeb a b then a = a - Npos b + Npos b else True.
 Proof.
   destruct a as [|a]; simpl; intros b; auto.
-  generalize (Pcompare_Eq_eq a b).
-  case_eq (Pcompare a b Eq); intros; auto.
-  rewrite H0; auto.
+  case Pos.compare_spec; intros; subst; auto.
   now rewrite Pminus_mask_diag.
-  destruct (Pminus_mask_Gt a b H) as [d [H2 [H3 _]]].
+  destruct (Pminus_mask_Gt a b (Pos.lt_gt _ _ H)) as [d [H2 [H3 _]]].
   rewrite H2. rewrite <- H3.
   simpl; f_equal; apply Pplus_comm.
 Qed.
@@ -96,7 +94,7 @@ rewrite Nplus_comm.
 generalize (NPgeb_correct (2*a+1) p). rewrite GE. intros <-.
 rewrite <- (Nmult_1_l (Npos p)). rewrite <- Nmult_plus_distr_r.
 destruct a; auto.
-red; simpl. apply Pcompare_eq_Lt; auto.
+red; simpl. apply Pcompare_Gt_Lt; auto.
 Qed.
 
 (* Proofs of specifications for these euclidean divisions. *)